Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Limit of $\frac{x}{x^2-1}$ Question
$$\lim_{x \rightarrow 1-}\frac{x}{x^2-1}$$
My attempt
$$\lim_{x \rightarrow 1-}\frac{x}{x^2-1}=\lim_{x \rightarrow 1-}x\lim_{x \rightarrow 1-}\frac{1}{x^2-1}=\lim_{x \rightarrow 1-}x\lim_{x \rightarrow 1-}\frac{x^2(\frac{1}{x^2})}{x^2(1-\frac{1}{x^2})}=\lim_{x \rightarrow 1-}x\lim_... | $$\lim_{x\to1^-}\frac x{x^2-1}$$
$$=\lim_{x\to1^-}\frac x{(x+1)(x-1)}$$
$$=\lim_{x\to1^-}\frac x{x+1}\cdot\frac1{x-1}$$
$$=\lim_{x\to1^-}\frac x{x+1}\cdot\lim_{x\to1^-}\frac1{x-1}$$
$$=\frac12\lim_{x\to1^-}\frac1{x-1}$$
$$\to-\infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Partial sums of harmonic series I've been given the following problem.
Prove that
$$\lim_{N\to\infty}\sum_{i=1}^N\frac1i=\infty.$$
In other words I need to prove that the partial sum of the harmonic series diverges. I know the integral test works in this case, but does anybody know of any other methods for showing t... | There are many ways to show this.
\begin{align}
& 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots \\[15pt]
= {} & 1 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots \\[8pt]
& \phantom{1} + \frac 1 2 + \frac 1 4 + \frac 1 6 + \cdots \\[15pt]
> {} & \phantom{{} + {}} \frac 1 2 + \frac 1 4 + \frac 1 6 + \cdots \\[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Some Pythagorean triples proofs I'm studying about number theory by myself. So I'm sorry if the question seems to be unclear.
I just want to know how to prove that one member of a Pythagorean triple is always divisible by 5 and that the area of an integer-sided right-angled triangle is always divisible by 6.
I searched... | We are considering integer solutions to
$$
a^2+b^2=c^2
$$
Let us consider the equation $a^2+b^2=c^2$ modulo $2$:
$$
\begin{array}{c|cc}
+&0^2&1^2\\
\hline
0^2&0^2&1^2\\
1^2&1^2&0^2
\end{array}
$$
we see that in all four cases at least one of the three numbers is zero modulo $2$. So at least one in the triple is divisi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Sum of the series $\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$
The sum of the series $$\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$$
$\bf{My\; Try::}$ We can write it as $\displaystyle \binom{n}{0} = $ Coefficient of $x^0$ in $(1+x)^n$
Similarly $\displaystyle \bino... | [Imported from a duplicate question]
Chebyshev polynomials of the second kind have the following representation:
$$ U_n(x)=\sum_{r\geq 0}\binom{n-r}{r}(-1)^r (2x)^{n-2r} \tag{1}$$
hence the wanted sum is just $U_n\left(\frac{1}{2}\right)$, and since $\frac{1}{2}=\cos\frac{\pi}{3}$,
$$ U_n\left(\frac{1}{2}\right) = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 5
} |
Describe all natural numbers $n$ for which $3^{n}-2^{n}$ is divisible by $5$. I need to describe all natural numbers $n$ for which $3^{n}-2^{n}$ is divisible by $5$.
After testing out some $n \in \mathbb{N}$, I came to the conclusion that $3^{n} - 2^{n}$ is divisible by $5$ iff $n$ is even - i.e., if $n$ is of the for... | $$3^{2k+1}-2^{2k+1}= 3(9^k)-2(4^k) \equiv 3(-1)^k-2(-1)^k=(-1)^k \mod 5$$
$$3^{2k}-2^{2k}= (9^k)-(4^k) \equiv (-1)^k-(-1)^k=0 \mod 5$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Probability and Conditional Probability Could someone please provide a worked solution to this question?
A box contains 5 amber, 7 blue and 9 green balls. Six of the balls are removed from the box at random and without replacement.
Find the probability that
(a) three out of the six balls are blue;
(b) four of the ball... | Well, for such combinatorical questions there are typically multiple different "counting schemes" to get to the same answer. Here is the thought process I would have to arrive at those. Hope it helps:
a) There are $\binom{5+7+9}{6}$ ways to choose the $6$ balls. There are $\binom{7}{3}$ ways to choose $3$ blue balls an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove $a {a+b \choose b}$ divides the lowest common multiple of $b+1, b+2, ..., b+a$ Prove that for $a, b \in \Bbb N^*$ we have :
$$a {a+b \choose b} | LCM(b+1, ..., b+a)$$
Is there a simple proof of this result that doesn't involve computing an integral? I know a trick exploiting $\int_{0}^{1} {x^n(1-x)^mdx}$ but I am... | First note that
$$a\binom{a+b}{b}=\frac{(a+b)!}{(a-1)!b!}$$
To prove that $a {a+b \choose b} | LCM(b+1, ..., b+a)$ it is enough to prove that for each prime $p$ we have
$$p^n | LCM(b+1, ..., b+a)$$
where $n$ is the power of $p$ in $a\binom{a+b}{b}$.
We know by Legendre formula that
$$n=\sum_{k}\lfloor \frac{a+b}{p^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
"Smart" solution for $7^{100}+11^{100} \pmod{13}$? I have the following exercise: "Find $7^{100}+11^{100} \pmod{13}$".
My long and boring solution is the following:
*
*I've found that $7^k \equiv 7^{k \pmod{12}} \pmod{13}$ by calculating $7^k \pmod{13} = 7 \cdot (7^{k-1} \pmod{13}) \pmod{13}$ from $k=1$ until the re... | Thank to the guys at the comment section, the improved solution is the following:
$$ 7^{100} + 11^{100}$$
$$= 7^{8 \cdot (13-1) + 4} + 11^{8 \cdot (13-1) + 4}$$
$$ \equiv 7^4 + 11^4$$
$$ \equiv (7^2)^2 + (-2)^4 $$
$$\equiv 10^2 + 16$$
$$\equiv 9 + 3 = 12 \pmod{13} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many tangent half-angle formulas are there? \begin{align}
\tan \frac{\alpha+\beta} 2 & = \frac{\sin\alpha+\sin\beta}{\cos\alpha + \cos\beta} \tag 1 \\[10pt]
\tan \left( \frac \pi 4 \pm \frac \alpha 2 \right) & = \sec\alpha \pm \tan\alpha \tag 2 \\[10pt]
\frac{1 + i\tan\frac\alpha2}{1-i\tan\frac\alpha2} & = e^{i\alp... | For $\alpha+\beta+\gamma=\pi$,
\begin{align}
\tan\tfrac\alpha2\,\tan\tfrac\beta2\tan\tfrac\gamma2
&=
\frac{\cos\alpha+\cos\beta+\cos\gamma-1}{\sin\alpha+\sin\beta+\sin\gamma}.
\end{align}
Edit
Some more
\begin{align}
\tan\tfrac\alpha2\,\tan\tfrac\beta2\tan\tfrac\gamma2
&=
\frac{2\sin\alpha\sin\beta\sin\gamma}{(\sin\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Given that $\sqrt[3]{1+9x} ≈ 1+3x+ax^2+bx^3$ for small values of $x$, find the values of the coefficients $a$ and $b$ Given that $\sqrt[3]{1+9x}\approx 1+3x+ax^2+bx^3$ for small values of $x$, find the values of the coefficients $a$ and $b$.
A question from CIE Maths Paper 33 O/N 2015. The answer is
$$a=-9$$
$$b=45$$
The ... | The binomial series
$$
(1+x)^\alpha=\sum\binom\alpha k x^k
$$
gives in your case
\begin{align}
(1+9x)^{\frac13}&=1+\frac13·9x+\frac12·\frac13·(\frac13-1)·(9x)^2\\&\qquad+\frac16·\frac13·(\frac13-1)·(\frac13-2)·(9x)^3\\&\qquad+\frac1{24}·\frac13·(\frac13-1)·(\frac13-2)·(\frac13-3)·(9x)^4+...\\
&=1+3x-9x^2+45x^3-270x^4+…... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Real values of $k$ for which $x^2+(k+1)x+k^2$ has one root double the other
For what real values of $k$ does $x^2+(k+1)x+k^2$ have one root double the other?
For a start, I found the range of $k$ which endows this equation with real roots:
$$-\frac13\le k\le1$$
| Let $a$ and $b$ be the roots. Since
$$
(x-a)(x-b)=x^2+(k+1)x+k^2\\
x^2-(a+b)x+ab=x^2+(k+1)x+k^2
$$
it follows (by comparing corresponding coefficients) that
$$
\begin{cases}
a+b=-(k+1) & (1)\\
ab=k^2 & (2)
\end{cases}
$$
If $a=2b$, then
$$
\begin{cases}
3b=-(k+1) & (3)\\
2b^2=k^2 & (4)
\end{cases}
$$
Flugging (3) ins... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
A circle inside an ellipse Consider an ellipse with semi-axes $a$ and $b$, taller than it is wide with a small circle of radius $r$ inside. Assume the circle falls to the lowest point possible while staying inside the ellipse.
If $2r\le a-c$ then the circle and ellipse will meet at a single point at the bottom. If $2r... | Never underestimate the power of cartesian coordinates
Consider the problem of finding the intersection points between ellipse and circle of radius $r$ centered in $(0,y_c)$, which leads to the system of equations
$$\begin{cases}\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1\\
x^2 + (y-y_c)^2 = r^2.\end{cases} $$
By substitution ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Find values of $a$ such that $\frac{x^2-x}{1-ax}$ attains all real values
Find values of $a$ such that
$$\frac{x^2-x}{1-ax}$$
attains all real values.
I first tried to find the range of the above by equating the above to some $y$. Solving I get $x^2+x(ay-1)-y=0$. How do I proceed?
| Suppose $f(x)=\frac{x^2-x}{1-ax}=k$ for some fixed constants $a,k$. Solving for $x$ we get
$$x^2-x=k(1-ax)=k-kax$$
$$x^2+(ka-1)x-k=0$$
$$x=\frac{1-ka\pm\sqrt{(ka-1)^2+4k}}2$$
For a chosen $a$, $f$ attains all real numbers if and only if the discriminant $(ka-1)^2+4k$ is non-negative for all $k$. Expanding:
$$(ak)^2-2ka... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Show that if $x + \frac1x = 1$, then $x^5 + \frac1{x^5} = 1$. Suppose $x + \frac{1}{x} = 1$.
Without first working out what $x$ is, show that
$x^5 + \frac{1}{x^5} = 1$ as well.
| Since $x + \frac{1}{x} = 1$ then
\begin{align}
1 &= 1^5 = \left( x + \frac{1}{x} \right)^{5} \\
&= \left( x^5 + \frac{1}{x^5} \right) + 5 \, \left( x^3 + \frac{1}{x^3} \right) + 10 \, \left( x + \frac{1}{x} \right) \\
&= \left( x^5 + \frac{1}{x^5} \right) + 5 \, \left(x + \frac{1}{x} \right)^3 - 15 \, \left( x + \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Prove that if $2b^2+1 = 3^b$ where $b$ is a positive integer, then $b \leq 2$
Prove that if $2b^2+1 = 3^b$ where $b$ is a positive integer, then $b \leq 2$.
Is there some way we can transform the equation in order to get the inequality? We have $2b^2 = 3^b-1$.
| Given: $ \qquad \displaystyle 2b^2+1 = 3^b \tag 1$
We see immediately $b=1$ and $b=2$ give equalities.
What if $b$ grows over $2$, so $b=2+c$? Let's first denote $ß=\lg3 $ and we know, $ß >1$ . With this we make the ansatz:
$$ 2(2+c)^2+1 = 3^{2+c} \tag 2$$
and develop by expanding and rearranging
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to solve $\sin(x) + 2\sqrt{2}\cos x =3$ How to solve $\sin(x) + 2\sqrt{2}\cos x=3$ ?
What is general method for doing these kind of questions?
Thanks
| If you make the following substitutions:
$$
\begin{cases}
X=\cos x\\
Y=\sin x
\end{cases}
$$
then your equations (remembering that $\sin^2 x+\cos^2x=1$) becomes
$$
\begin{cases}
X^2+Y^2=1\\
Y+2\sqrt{2}X=3
\end{cases}
$$
which is the intersection between a circle and a straight line.
Solving it:
$$
\begin{cases}
Y=3-2\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Given $P(x)=x^{4}-4x^{3}+12x^{2}-24x+24,$ then $P(x)=|P(x)|$ for all real $x$ $$P(x)=x^{4}-4x^{3}+12x^{2}-24x+24$$
I'd like to prove that $P(x)=|P(x)|$. I don't know where to begin. What would be the first step?
| As the other answers pointed out, you need to prove that the polynomial is positive for all values of $x$. A very common way to do it is to express it as a sum of squares.
$$P(x)=x^4−4x^3+12x^2−24^x+24$$
The problems here are the two terms where X is at an odd power: $-4x^3$ and $-24x$
Trying to fit them into squares i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 10,
"answer_id": 6
} |
Beta function with different integral limits The beta function is
$$B(x,y) = 2\int_{0}^{\frac{\pi}{2}} \cos^{2x-1}\theta \sin^{2y-1}\theta d\theta$$
for positive values of $x$ and $y$.
How can this integral be used to evaluate other limits of integration?
For example,
$$ \int_{0}^{\pi} \sqrt{\sin\theta} \, d\theta$$
| Consider:
\begin{align}
\int_{0}^{\pi} \sqrt{\sin\theta} \, d\theta &= \int_{0}^{\pi/2} \sqrt{\sin\theta} \, d\theta + \int_{\pi/2}^{\pi} \sqrt{\sin\theta} \, d\theta \\
&= \int_{0}^{\pi/2} \sqrt{\sin\theta} \, d\theta + \int_{0}^{\pi/2} \sqrt{\sin(\theta + \pi/2)} \, d\theta \\
&= \int_{0}^{\pi/2} \sqrt{\sin\theta} \,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding an unknown coefficient of a polynomial given a factor Q:Find the value of $a$ given that $x^2+1$ is a factor of $x^4-3x^3+3x^2+ax+2$
No idea where to start, I was going to use the factor theorem but it didn't work out.
Question from year 10 Cambridge maths textbook
| \begin{align}
x^4-3x^3+3x^2+ax+2 &= (x^4 + x^2) - 3 \, (x^3 + x) + 2 \, (x^2 + 1) + (a+3) \, x \\
&= (x^2 + 1) \, \left( x^2 - 3 \, x + 2 + \frac{(a+3) \, x}{x^2 + 1} \right)
\end{align}
In order for this to be factored then it is required that $a = -3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Does this trigonometric pattern continue for all primes $p=6m+1$? (Revised from its original form.) Consider primes $p=6m+1$, and $p>13$:
$p=19=3\times\color{blue}{6}+1$:
Let $\beta = 2\pi/19.\,$ A root of $x^3+x^2-\color{blue}{6}x-7=0$ is $$x=2\big(\cos(2\beta)+\cos(3\beta)+\cos(5\beta)\big)$$
Note that $2+3=5$, o... | This situation can be summarized as follows (I take the case p = 19 as an example):
Let $\zeta$ be a $19$-th root of unity then we can form the field of cyclotomic numbers $\mathbb{Q}(\zeta)$. The galois group of this extension (of $\mathbb{Q}$) is the cyclic group $C_{18}$ of order $18$. In order to construct a sub-e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\lim ( \sqrt{n^2+n}-n) = \frac{1}{2}$ Here's what I have so far:
Given $\epsilon > 0$, we want to find N such that $\sqrt{n^2+n}-n < \epsilon$ for all $n>N$. And so:
$( \sqrt{n^2+n}-n-\frac{1}{2}) \cdot \frac{\sqrt{n^2+n}-(n+\frac{1}{2})}{\sqrt{n^2+n}-(n+\frac{1}{2})}$
$= \frac{(n^2+n)-(n+\frac{1}{2})}{\sq... | For each $\varepsilon>0$ you want to find $N$ such that, for $n>N$,
$$
\left|\sqrt{n^2+n}-n-\frac{1}{2}\right|<\varepsilon
$$
that is,
$$
-\varepsilon+\frac{1}{2}<\sqrt{n^2+n}-n<\varepsilon+\frac{1}{2}
$$
It is not restrictive to assume $0<\varepsilon<1/2$.
The inequality $\sqrt{n^2+n}-n>A$, for $A<1/2$, is the same as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find all values of $ r $ such that $ a_n = r^n $ for $ n \in \mathbb{Z}_{\ge2} $ in a recurrence relation I am trying to solve the following problem:
Given the following recurrence relation:
\begin{equation}
a_n = 6a_{n-1} - 8a_{n-2} \ \text{ for } \ n \in \mathbb{Z}_{\ge2}
\end{equation}
Find all values of $ r $ so th... | Both roots give solutions: $r=4$ for $a_n = 4^n$ ($C=0, D=1$) and $r=2$ for $a_n = 2^n$ ($C=1, D=0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
The sequence $x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. I have problem
Let $m \ge 2 -$ fixed positive integer. The sequence of non-negative real numbers $\{x_n\}_{n=1}^{\infty}$ is that for all $n\in \mathbb N$
$$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$
Prove that this sequence has a limit.
H... | Note that, given the unilateral z-transform
$$
A(z) = \sum\limits_{0\, \leqslant \,n} {a_{\,n} \,z^{\,n} } \quad \left| {\;a_{\,n < 0} = 0} \right.
$$
then the Moving Window Sum has a z-transform:
$$
\begin{gathered}
W(A(z),h) = \sum\limits_{0\, \leqslant \,n} {\left( {a_{\,n - h + 1} + \cdots + a_{\,n - 1} + a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$ I stumbled upon the interesting definite integral
\begin{equation}
\int\limits_0^1 \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2
\end{equation}
Here is my proof of this result.
Let $u=\sin^{-1}(x)$ then integrate by parts,
\begin{align}
\int \frac{\sin... | \begin{array}{r}
\displaystyle \int_{0}^{1} \frac{\arcsin x}{x} d x=& \int_{0}^{1} \arcsin x d(\ln x)
\stackrel{IBP}{=} \displaystyle -\int_{0}^{1} \frac{\ln x}{\sqrt{1-x^{2}}} d x
\end{array}
Let $\displaystyle I(a)=\int_{0}^{1} \frac{x^{a}}{\sqrt{1-x^{2}}} d x$ and $x=\sin \theta$, then our integral becomes $$
\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 3
} |
Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.
Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.
The problem can be solved by putting every element from $\mathbb{Z}_{11}$ and checking if there is any root.
$0^2 + 0 + 4 = 4 \\
1^2 + 1 + 4 = 6 \\
2^2 + 2 + 4 = 10 \\
3^2 + 3 + 4 = 5 \\
4^2 + 4 + 4 = ... | By completing the square we get
$$x^2+x+4=(x+6)^2+1.$$
So the original question reduces to the question if $-1$ is a square in $\mathbb Z_{11}$. This question can be answered with no using the first supplement for the law of quadratic reciprocity since $11 \equiv 3 \pmod 4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1993023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove that $x_n = 1 - \sum\limits^{n}_{i=1} \frac{1}{n+i}$ converges
Let $(x_n)^{\infty}_{n=1}$ be the sequence $\displaystyle{x_n = 1 - \sum^{n}_{i=1} \frac{1}{n+i}}$. Prove that $(x_n)^{\infty}_{n=1}$ converges.
I know the sequence is decreasing and bounded below, hence will converge (Monotone Convergence Theorem),... | If
$x_n = 1 - \sum^{n}_{i=1} \frac{1}{n+i}
$,
then
$\begin{array}\\
x_{n+1}-x_n
&=\left(1 - \sum^{n+1}_{i=1} \frac{1}{n+1+i}\right)-\left(1 - \sum^{n}_{i=1} \frac{1}{n+i}\right)\\
&=\sum^{n}_{i=1} \frac{1}{n+i}-\sum^{n+1}_{i=1} \frac{1}{n+1+i}\\
&=\sum^{n}_{i=1} \frac{1}{n+i}-\sum^{n+2}_{i=2} \frac{1}{n+i}\\
&=\frac1{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find a shortest distance from a point to an instersection of two implicit surfaces I have a problem where I need to find a point on a curve that is defined by two implicit surfaces. This point must be at the shortest distance from a reference point that is located somewhere in space.
These two surfaces are a cone-like ... | $$
\left \{
\begin{align*}
x^2+y^2+z^2 &= 1 \\
2(x^2+y^2) &= (z+1)^{2}
\end{align*}
\right.$$
The cone here has a nice symmetry, we can eliminate $x$ and $y$ first:
\begin{align*}
2(1-z^2) &= (z+1)^2 \\
3z^2+2z-1 &=0 \\
(z+1)(3z-1) &= 0 \\
z &= -1 \quad \text{or} \quad \frac{1}{3}
\end{align*}
When ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove: $\log_{2}{3} < \log_{3}{6}$ How should I prove that $\log_{2}{3} < \log_{3}{6}$?
I tried something like this: $2^{\log_{2}{3}}< 2^{\log_{3}{6}}$, $3<6^{\log_{3}{2}}$, $\log_{6}{3}<\log_{6}{6^{\log_{3}{2}}}=\log_{3}{2}$, $\frac{1}{\log_{3}{6}}< \log_{3}{2}$. $\frac{1}{1+\log_{3}{2}}<\log_{3}{2}$, but still nothi... | This is essentially the same answer as barak manos's and Djura Marinkov's, mostly just presented in a different fashion, as one long string of self-explanatory equalities and inequalities:
$$\begin{align}
5\log_23&=\log_2243\\
&\lt\log_2256\\
&=8\\
&=5+\log_327\\
&\lt5+\log_332\\
&=5\log_33+5\log_32\\
&=5\log_36
\end{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Hardcore integral with absolute value Is it possible to solve this integral?
$$\int_0^1\int_0^1\frac{{(y-y_1-\frac{(x-x_1)(y_2-y_1)}{x_2-x_1})(y-y_1-\frac{(x-x_1)(y_3-y_1)}{x_3-x_1})(y-y_3-\frac{(x-x_3)(y_2-y_3)}{x_2-x_3})}}{|{(y-y_1-\frac{(x-x_1)(y_2-y_1)}{x_2-x_1})(y-y_1-\frac{(x-x_1)(y_3-y_1)}{x_3-x_1})(y-y_3-\frac{... | Label the triangle vertices $v_{1} = (x_{1}, y_{1})$, $v_{2} = (x_{2}, y_{2})$, $v_{3} = (x_{3}, y_{3})$, counterclockwise (say), so that the line $\ell_{ij} = \overline{v_{i} v_{j}}$ divides the square into two regions; call the "right-hand" region (not containing the central triangle) $A_{ij}$. Let $a_{ij}$ denote th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Partition Proving Let $A=\{2,6,10,14,\ldots\}$ be the set of integers that are twice an odd number.
Prove that, for every positive integer $n$, the number of partitions of $n$ in which no odd number appears more than once is equal to the number of partitions of $n$ containing no element of $A$.
For example, for $n=6$, ... | Let $f(n)$ be the partitions of $n$ with at most one occurrence of each odd number, and $F(z)=\sum_{n} f(n)z^n$.
Then $$\begin{align}F(z) &= (1+z)(1+z^2+z^4+\cdots)(1+z^3)(1+z^4+z^8+\cdots)\cdots \\&=\frac{(1+z)(1+z^3)(1+z^5)\cdots}{(1-z^2)(1-z^4)(1-z^6)\cdots}
\end{align}$$
Now pair off the terms $\frac{1+z^{2k+1}}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Calculating limit of $\lim_{x\to\infty}\dfrac{\sqrt{x+1}-2\sqrt{x+2}+\sqrt{x}}{\sqrt{x+2}-2\sqrt{x}+\sqrt{x-4}}$ As the title says we want to calculate:
$$\lim_{x\to\infty}\dfrac{\sqrt{x+1}-2\sqrt{x+2}+\sqrt{x}}{\sqrt{x+2}-2\sqrt{x}+\sqrt{x-4}}$$
By multiplying nominator and denominator in their conjugates
$=\lim_{x\t... | You can write the fraction as $$\frac{\sqrt{x+1}-2\sqrt{x+2}+\sqrt{x}}{\sqrt{x+2}-2\sqrt{x}+\sqrt{x-4}}=\frac{(\sqrt{x+1}-\sqrt{x+2})+(\sqrt{x}-\sqrt{x+2})}{(\sqrt{x+2}-\sqrt{x})+(\sqrt{x-4}-\sqrt{x})}=\frac{-\frac{1}{\sqrt{x+2}+\sqrt{x+1}}-\frac{2}{\sqrt{x}+\sqrt{x+2}}}{\frac{2}{\sqrt{x+2}+\sqrt{x}}-\frac{4}{\sqrt{x-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Proving $∠CAD = 90◦$ In a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ intersect at
$X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$.
If $X$ is the incentre of triangle $ABY$ , show that $∠CAD = 90^{\circ}$.
| First we show that point $Y$ lies on the edges $CD$. Look at quadrilateral $XCYD$. We will prove that $\angle \, CYD = 180^{\circ}$.
*
*$\angle\, XDY = \angle \, XAY = \alpha$ as inscribed in a circle.
*$\angle \, BAC = \angle \, BAX = \angle \, XAY = \alpha$ since $AC$ passes through the incenter $X$ of triangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
simplify $\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$ simplify $\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$.
1.$90^{\frac{3}{2}}$
2.$106\sqrt{41}$
3.$4\sqrt{41}$
4.$504$
5.$508$
My attempt:I do like this but I didn't get any of those five.
$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}={\sqrt{45+4\s... | Is this the nested radical formula?
$\sqrt{45 \pm 4\sqrt{41}} = a \pm b\sqrt{41}$
$45 \pm 4\sqrt{41} = (a^2 + 41b^2) \pm 2ab\sqrt{41}$
$a^2 + 41b^ = 45; 2ab = 4 \implies a=2;b = 1$
So $\sqrt{45 \pm 4\sqrt{41}} = |2 \pm \sqrt{41}|= \pm 2 + \sqrt{41}$
Plugging that into: $=(\sqrt{45+4\sqrt{41}}-\sqrt{45-4\sqrt{41}})(45+4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Evaluate the integral $\int \:\frac{\sqrt{x^2+2\cdot\:\:x+2}}{x}dx$ $$\int \:\frac{\sqrt{x^2+2\cdot\:\:x+2}}{x}dx$$
I have tried to form a square above i also tried to get the x below under the root
but got nothing
| Rationalise the numerator.
$\displaystyle \int \frac{\sqrt{x^2+2x+2}}{x} \, \text{d}x = \int \frac{x^2+2x+2}{x\sqrt{x^2+2x+2}}\, \text{d}x$
Split apart as follows:
$\displaystyle \int \frac{x^2+2x+2}{x\sqrt{x^2+2x+2}}\, \text{d}x = \int \frac{x+1}{\sqrt{x^2+2x+2}} \, \text{d}x + \int \frac{\text{d}x}{\sqrt{x^2+2x+2}} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Find and prove the limit $\lim_{n \to \infty}\frac{2^{n+2}+3^{n+3}}{2^n+3^n}$ I'm trying to find and prove the limit $$\lim_{n \to \infty}\frac{2^{n+2}+3^{n+3}}{2^n+3^n}$$
which according to WolframAlpha happens to be $27$.
The furthest I could get is
$$\frac{2^{n+2}+3^{n+3}}{2^n+3^n} = 4\left(1 + \frac{3^{n}}{2^n+3^n}... | $$\frac{2^{n+2}+3^{n+3}}{2^n+3^n}=\frac{(4)2^{n}+(27)3^{n}}{2^n+3^n}=\frac{(4)2^{n}/3^n+(27)1}{2^n/3^n+1}$$
so the limit is $27$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2005247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Infimum of complex expression on the unit disc in the complex plane Suppose $|\lambda|<1, Re\lambda < 0, Im\lambda > 0$. What is the largest lower bound independent of $\lambda$ on the expression
$$\frac{1+|\lambda|^2}{2}+Re\lambda - \frac{2\pi}{n+2}Im\lambda$$ where $n$ is a positive integer? If you like, stipulate t... | This is equivalent to the bivariate real optimization problem:
\begin{array}{ll}
\text{minimize} & \frac{1+x^2+y^2}{2}+x-\frac{2\pi}{n+2} y \\
\text{subject to} & x^2 + y^2 \leq 1
\end{array}
To solve, we build the Lagrangian
$$L(x,y,z) = \frac{1+x^2+y^2}{2}+x-\frac{2\pi}{n+2} y - z(1 - x^2 - y^2)$$
where $z \geq 0$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2005898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Induction divisibility question Q. Prove by induction that $2^{3n-1} + 5(3^n)$ is divisible by $11$ for any even number $n$, where $n$ is an element of natural numbers.
What is have so far:
(base case): $p(2) = 77$, $77/11 = 7$. so base case holds
$p(k) = 2^{3k-1} +5(3^k) $
$p(k+2) = 2^{3k+5} + 5(3^{k+2}) $
$p(k+2) = 2... | You want to relate $p(k+2)$ back to $p(k)$, and you've gotten most of the way there (though you could've gone straight from $p(k+2)=2^{3k+5}+5(3^{k+2})$ to $p(k+2)=2^{3k-1}2^6+5(3^k)(3^2)$).
It would be nice if instead of $2^{3k-1}2^6+5(3^k)(3^2)$ you had something like $2^{3k-1}2^6+5(3^k)(2^6)$ because then you could ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2007491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Solving simultaneous equations involving a quadratic I have the question
Solve the simultaneous equation pair
$$x^2 + y^2 = 25\tag1$$
$$2x - y = 5\tag2$$
I have found the value of $y$ from the second equation which is $2x-5$ and substituted this into the first equations $y$ value.
I get $x^2 + (2x -5)^2 = 25$
When I... | $$
\begin{cases}
\text{x}^2+\text{y}^2=25\\
2\text{x}-\text{y}=5
\end{cases}\space\Longleftrightarrow\space
\begin{cases}
\text{x}^2+\left(2\text{x}-5\right)^2=25\\
\text{y}=2\text{x}-5
\end{cases}
$$
Solving:
$$\text{x}^2+\left(2\text{x}-5\right)^2=25$$
Gives us $x=0$ or $x=4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2007864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Show that $b = ln(\sqrt[3]{\frac{18}{27-k}})$ for $9 = \frac{k}{3-2e^{-b3}}$ I did:
$$9 = \frac{k}{3-2e^{-b3}} \Leftrightarrow \frac{k}{9} = 3-2e^{b3} \Leftrightarrow \frac{k}{9}-\frac{3}{1} = -2e^{-3b} \Leftrightarrow \frac{\frac{k-27}{9}}{\frac{-2}{1}} \Leftrightarrow -\frac{k-27}{18} = \frac{1}{e^{2b}} \Leftrightarr... | you are there.
$$
\frac{1}{a-b} = -\frac{1}{b-a}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to find x and y coordinates based on the given distance? The problem says: Find the point (coordinates $(x,y)=~?$) which is symmetrical to the point $(4,-2) $ considering the given equation $y=2x-3$
I have found the perpendicular line-slope $y=-~\frac{1}{2}x$ and the
intersection point which is shown in the graph... | I think you're over-thinking it. When you have the intersect point and another point, you just duplicate the difference to get the point on the other side.
$(\frac{6}{5}, -\frac{3}{5}) - (4, -2)$ is $(-\frac{14}{5}, \frac{7}{5})$
That is what you need to add to the point to get to your intersect point, so either add t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
What is the mistake? $$1=1$$
$$\Rightarrow\frac{-1}{1}=\frac{1}{-1}$$
$$\Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$
$$\Rightarrow\frac{i}{1}=\frac{1}{i}$$
$$\Rightarrow\frac{i}{2}=\frac{1}{2i}$$
$$\Rightarrow\frac{i}{2}+\frac{3}{2i} = \frac{1}{2i} +\frac{3}{2i}$$
$$\Rightarrow i(\frac{i}{2}+\frac{3}{2i} ) = ... | The trick is assuming that
$$\sqrt{\frac{-1}{1}} = \sqrt{\frac{1}{-1}}\Rightarrow
\frac{\sqrt{-1}}{\sqrt{1}} = \frac{\sqrt{1}}{\sqrt{-1}}$$
which is not true because by definition of (complex) square root $-1=\sqrt{-1}\cdot \sqrt{-1} \not=\sqrt{1}\cdot \sqrt{1}=1$.
On the other hand, note that if $a,b,c,d$ are real PO... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2013261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Maximum value of algebraic expression
If $2a+3b+4c=100$ and $a,b,c\in \mathbb{N}\;,$ Then $\max(2a^2+5b^2+8c^2)$
$\bf{My\; Try::}$ We can write it as $\displaystyle \frac{a}{50}+\frac{b}{\frac{100}{3}}+\frac{c}{25} = 1$
So it represent a plane which cut $x,y,z$ axis at $\displaystyle A(50,0,0)$ and $\displaystyle B\l... | First note that $2a+3b + 4c=100 \implies b \equiv 0 \pmod 2$ and $2a+c \equiv 1 \pmod 3$.
As $b$ is even, we may simplify a bit by letting $b = 2k$, then $a+3k+2c = 50$ and we maximize $V = a^2+10k^2+4c^2$.
As $50^2 = a^2+9k^2+4c^2 + 2(3ak+6kc+2ca)$, we have
$$V= k^2+50^2 - 2(3ak+6kc+2ca) = (k-(3a+6c))^2+50^2 - 4ca ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2014791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the value of $\lim\limits_{n \to \infty}n\left(\left(\int_0^1\frac{1}{1+x^n}\,\mathrm{d}x\right)^n-\frac{1}{2}\right)$
Find the value of the limit $$\lim_{n \to \infty}n\left(\left(\int_0^1 \frac{1}{1+x^n}\,\mathrm{d}x\right)^n-\frac{1}{2}\right)$$
I can't solve the integral $\int_0^1 \mathrm{\frac{1}{1+x^n}}\,\... | Hint. If one knows the digamma function $\psi=\Gamma'/\Gamma$, one may write, as $n \to \infty$,
$$
\begin{align}
\int_0^1\frac1{1+x^n}\:dx&=\int_0^1\frac{1-x^n}{1-x^{2n}}\:dx
\\\\&\stackrel{u=x^{2n}}=\frac1{2n}\int_0^1\frac{(1-u^{1/2})u^{1/(2n)-1}}{1-u}\:du
\\\\&=\frac1{2n}\left[\psi\left(\frac1{2n}+\frac12 \right)-\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2015233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
another trig limit without L'Hospital? $$\lim_{x\to \pi/3 }\dfrac{1-2\cos\left(x\right)}{{\pi}-3x}$$
Here's what I tried:
${\pi}-3x=y$,
$\dfrac{{\pi}-y}{3}=x$
$$\lim_{y\to\ 0} \dfrac{1-2\cos\left(\frac{{\pi}-y}{3}\right)}{y}$$
...
| Here is a proof without L'Hôpital's rule or derivatives. The two basic trigonometric limits are
\begin{align*}
\lim_{\theta\to 0} \frac{\sin\theta}{\theta} &= 1\\
\lim_{\theta\to 0} \frac{1-\cos\theta}{\theta} &= 0
\end{align*}
The first can be shown geometrically, and the second by an algebraic manipulation o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Given ${[1+ {(1+ x)}^{1/2}]×\tan(x) = \left[1+ {(1- x)}^{1/2}\right]}$, find $\sin 4 x$. If
$${[1+ {(1+ x)}^{1/2}]×\tan(x) = \left[1+ {(1- x)}^{1/2}\right]}$$
then find the value of $\sin(4x)$.
The options given are:
a) $x$
b) $4x$
c) $2x$
I tried applying many trigo identities but none of them is working and the radi... | Let: $$t = \tan(x) = \frac{1+\sqrt{1-x}}{1+\sqrt{1+x}}$$
Using the double-angle sine and half-angle tangent formulas:
$$
\sin(4x) \;=\; 2\;\sin(2x)\;\cos(2x) \;=\; 2 \cdot \frac{2 t}{1 + t^2} \cdot \frac{1-t^2}{1+t^2} \;=\; \frac{4t(1-t^2)}{(1+t^2)^2}
$$
Substituting back $t$ in terms of $x$ and simplifying:
$$
\requir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2018782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How many squares are of the form $x^3+7x^2-x$?
How many squares are of the form $x^3+7x^2-x$, where $x$ is a positive integer, that are not multiples of $16$?
Since $x^3+7x^2-x = x(x^2+7x-1)$, we need both $x$ and $x^2+7x-1$ to be perfect squares and so let $x = k^2$, for some $k \in \mathbb{Z}^+$ . Then we need $(k^... | @ovi Why both $x$ and $ x^2+7x-1$ must be a square? Because they are coprime.
From $x^2+7x-1=k^2$ we get $x^2+7x-1-k^2=0$ and the discriminant is $53+4k^2 = m^2 $. From here $53=(m-2k)(m +2k)$ and $53$ is a prime number. It follows $m+2k = 53, m-2k = 1$ and $m=27, k=13$. Therefore $x^2+7x-170=0$ and, from here, $x=10$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Diophantine equation $x^4+5y^4=z^4$ I am trying to find all positive integer solution $(x,y,z)$ of equation $x^4+5y^4=z^4$.
Here I fould: $(x,y,z)=(1,2,3)$ and $(d,2d,3d)$. I try to prove if $(x,y)=1$ then $(1,2,3)$ is the unique solution of equation. Could anyone help me for this question?
| I experimentally searched the possibility of more principal solutions for this equation.
$x^4 +5y^4=z^4$
$z^4≡ x^4\mod y^4$⇒ $x^4<y^4$ or $x<y$
$z^4≡ x^4\mod 5$ ⇒ $x^4 < 5$; the only possible number for x is $1$, that is $x=1$⇒ $(y=2)$ and $z=3$ are the principal solutions.
Also we may write:
$(z^2+x^2)(z+x)(z-x)=5y^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
} |
Taylor expansion for $\frac{1}{x^2 + 2x + 2}$ I'm trying to find the Taylor expansion of the function:
$$ f(x) = \frac{1}{x^2 + 2x + 2} $$
about the point $ x = 0 $. I have worked out the terms up to the fourth derivative, which was very tedious. I found:
$$ f(x) = \frac{1}{2} - \frac{1}{2} x + \frac{1}{4} x^2 + 0 x^3 ... | $$ f(x) = \frac{1}{x^2 + 2x + 2} = a_0 + a_1 x + a_2 x^2 + \cdots $$
\begin{align*}
&& (x^2 + 2x + 2)f(x) &= 1 \quad ({}+ 0 x + 0x^2 + 0x^3 + \cdots) \\
&\implies& 2a_0 &= 1 \\
&& 2a_1 + 2a_0 &= 0 \\
&& 2a_2 + 2a_1 + a_0 &= 0 \\
&& 2a_3 + 2a_2 + a_1 &= 0 \\
&& 2a_4 + 2a_3 + a_2 &= 0 \\
&&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 5
} |
A limit without invoking L'Hopital: $\lim_{x \to 0} \frac{x \cos x - \sin x}{x^2}$ The following limit
$$\ell=\lim_{x \rightarrow 0} \frac{x \cos x - \sin x}{x^2}$$
is a nice candidate for L'Hopital's Rule. This was given at a school before L'Hopital's Rule was covered. I wonder how we can skip the rule and use basic l... | The function $\frac{x\cos(x)-\sin(x)}{x^2}$ is odd, so we only need to look at $0\lt x\lt\frac\pi2$. As shown in this answer, $0\le\sin(x)\le x\le\tan(x)$. Furthermore, $x\cos(x)-\sin(x)=(x-\tan(x))\cos(x)\le0$. So we have
$$
\begin{align}
0
&\ge\color{#C00000}{\frac{x\cos(x)-\sin(x)}{x^2}}\\
&\ge\frac{\sin(x)(\cos(x)-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2027117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Arithmetic proof of absolute value function of complex numbers I am looking for the arithmetic proof that:
$ |z| = \sqrt{(x^2 + y^2)} $ where $ z = x + i y $
Previously I assumed squaring a function then square rooting it would be analogous to the absolute value function (modulus) but it seems not to be the case i... | If $|x|=\sqrt {x^2} $ for $x\in \mathbb R $ is not a definition but a result.
$\mathbb R\subset \mathbb C$ and $|x|$ is still defined to be $|x|=|x+0i|=\sqrt {x^2+0^2}=\sqrt{x^2} $.
The definition is $|z|=\sqrt {\text {Re} (z)^2+\text {Im}(z)^2} $ whether $z$ is purely real, complex, or purely imaginary.
In general $|z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
What will be the 49th derivative? Let be the function $f=(x^3+3x) \cdot sin(x)$
$f'(x)=\left(3x^2+3\right)\sin\left(x\right)+\left(x^3+3x\right)\cos\left(x\right)$
$f''(x)=\left(3x-x^3\right)\sin\left(x\right)+\left(6x^2+6\right)\cos\left(x\right)$
$f'''(x)=\left(-9x^2-3\right)\sin\left(x\right)+\left(15x-x^3\right)\co... | So the general Leibniz rule somewhat is similar to the binomial theorem. Let $u,v$ be n-th differentiable functions then $(uv)^{(n)}=\sum_{k=0}^{n}{n \choose k}u^{(k)}v^{(n-k)}$ where the exponents denote the $kth$ derivative.
When now note that the fourth derivative and higher of $x^3+3x$ will be zero so we only need ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Given $a^2+b^2=2$ prove $a+b\le2$
*
*Given $a^2+b^2=2$ prove $a+b\le2$
*Given $a+b=2$ prove $a^4+b^4\ge2$
I was trying to prove these using the fact that we know $a^2+b^2\ge2ab$ but not sure where to start.
| By $1$-d version of polarization identity $(a+b)^2 + (a-b)^2 = 2(a^2+b^2)$,
*
*If $a^2 + b^2 = 2$, then
$$(a+b)^2 = 2(a^2+b^2) - (a-b)^2\le 2(a^2+b^2) = 4 \quad\implies\quad a+b \le |a+b| \le 2$$
*If $a + b = 2$, then
$$
8(a^4+b^4) = 4((a^2+b^2)^2 + (a^2-b^2)^2)\\
\ge 4(a^2+b^2)^2 = (2(a^2+b^2))^2 = ((a+b)^2 + (a-b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How is $2(\cos^2 x - \sin^2 x)(\sin x - \cos x) = -2(\cos x - \sin x)^2(\cos x + \sin x)$? I was reading the answer of a proof and in it the answer converted this:
$$2(\cos^2(x) - \sin^2(x))(\sin(x) - \cos(x))$$
to this
$$-2(\cos(x) - \sin(x))^2(\cos(x) + \sin(x))$$
can someone explain what happened and how did it end ... | We just use $a^2-b^2=(a-b)(a+b)$:
\begin{equation}
\begin{split}
2(\cos^2x - \sin^2 x)(\sin x - \cos x)&=-2(\cos^2 x - \sin^2 x)(\cos x - \sin x)\\
&=-2(\cos x - \sin x)(\cos x + \sin x)(\cos x - \sin x)\\&=-2(\cos x - \sin x)^2(\cos x + \sin x)
\end{split}
\end{equation}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2039882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem with proof of derivate of $arcsecx$ Look at this proof for derivative of $arcsecx $
$$y=arcsecx \Rightarrow secy=x$$
We have:
$$(secy) \prime =secytany$$
We take derivations from both parts with respect to $x $
$$y \prime = \frac{1}{secytany} = \frac{1}{secy \sqrt{sec^2y-1} } \Rightarrow y \prime = \frac{1}{... | Hint:
$$
\tan y= \frac{\sin y}{\cos y}=\frac{\sqrt{1-\cos^2 y}}{\cos y}=\sec y \sqrt{1-\frac{1}{\sec ^2 y}} =
$$
$$
= \sec y \sqrt{\frac{\sec^2 y -1}{\sec^2 y}}=\frac{\sec y}{\color{red}{|\sec y|}}\sqrt{\sec^2 y-1}=\color{red}{\mbox{sign}(\sec y)}\sqrt{\sec^2 y-1}
$$
In simple words: the problem is that $\sqrt {a^2}=|a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A nice pattern for the regularized beta function $I(\alpha^2,\frac{1}{4},\frac{1}{2})=\frac{1}{2^n}\ $? In this post, the problem was given integer/rational $N$, to solve for algebraic number $z$ in the equation,
$$\begin{aligned}\frac{1}{N}
&=I\left(z^2;\ a,b\right)\\[1.5mm]
&= \frac{B\left(z^2;\ a,b\right)}{B\left(a,... | We have
$$
B\left(z;\ \tfrac14,\tfrac12\right)=4\sqrt[4]{z}\, _2F_1\left(\frac{1}{2},\frac{1}{4};\frac{5}{4};z\right).
$$
By formula 2.1.15 from Erdelyi, "Higher transcendental functions", vol.I
$$
_2F_1\left(\frac{1}{2},\frac{1}{4};\frac{5}{4};z\right)=\sqrt{\frac{1}{1-z}} \, _2F_1\left(\frac{1}{2},\frac{1}{4};\frac{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
parts of cubic equation Below there is a solved question from a book .
I could not understand how they got $S_{n+3}$ = $S_{n+1}$ - $S_n$
| All the exercise is centered onto Newton's Identities.
In particular, concerning your question, we have that putting
$$
x^3 - x + 1 = e_0 x^3 - e_1 x^2 + e_2 x - e_3 = 0
$$
then
$$
\begin{gathered}
e_0 = 1 \hfill \\
e_1 = 0 = S_1 \hfill \\
e_2 = - 1 = \frac{1}
{2}\left( {e_1 S_1 - S_2 } \right) = - \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
how to show $a_{n}=[\frac{(2n)!!}{(2n-1)!!}]^2 \frac{1}{2n+1}$ converges? Question: $\displaystyle{a_{n} =
\left[{\left(2n\right)!! \over \left(2n - 1\right)!!}\,\right]^{2}
{1 \over 2n + 1}\,,\quad\mbox{prove}\ a_{n}}$ converges.
My thought: I want to prove {$a_{n}$} is an increasing sequence and it has an upper bound... | Consider the sequence $(b_n)$ defined by
$$b_n = \frac{2n+2}{2n+1}\cdot a_n.$$
Then $(b_n)$ is decreasing
\begin{align}
\frac{b_{n+1}}{b_n} &= \frac{\frac{2n+4}{2n+3}}{\frac{2n+2}{2n+1}}\cdot\frac{a_{n+1}}{a_n} \\
&= \frac{(2n+4)(2n+1)}{(2n+2)(2n+3)}\cdot\frac{(2n+2)^2}{(2n+1)(2n+3)} \\
&= \frac{(2n+4)(2n+2)}{(2n+3)^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Evalute $\int\frac{x^5}{(36x^2+1)^{3/2}}dx$ by trig sub? I'm stuck trying to evaluate the indefinite integral $\int\frac{x^5}{(36x^2+1)^{3/2}}dx$. It looks like it might be solvable by trig substitution, where $tan^2\theta+1=sec^2\theta$. That strategy seemed to payoff until I eliminated the square root. When I've work... | You are almost there.
$$\int\frac{\sin^5(\theta)}{\cos^4(\theta)}d\theta=\int\frac{\sin(\theta)(1-\cos^2(\theta))^2}{\cos^4(\theta)}d\theta=-\int\frac{1-2u^2+u^4}{u^4}du\\
=-\int u^{-4}-2u^2+1 du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2050611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to solve system of equations involving square roots How to solve the following system of equations? I've tried some basic techniques like adding/substracting and squaring but with no effect.
$$
\left\{
\begin{array}{c}
\sqrt{1 + x_1} + \sqrt{1 + x_2} + \sqrt{1 + x_3} + \sqrt{1 + x_4} = 2\sqrt{5} \\
\sqrt{1 - x_1}... | Let $\mathbf{r}_k=
\begin{pmatrix}
\sqrt{1+x_{k}} \\ \sqrt{1-x_{k}}
\end{pmatrix}$, then $\displaystyle \sum_{k=1}^{4} \mathbf{r}_k=
\begin{pmatrix}
2\sqrt{5} \\ 2\sqrt{3}
\end{pmatrix}$
Now,
\begin{align*}
\left| \sum_{k=1}^{4} \mathbf{r}_k \right| &=2\sqrt{5+3} \\
&= 4\sqrt{2} \\
\sum_{k=1}^{4} |\mathbf{r}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
lim 1/(x-3) (epsilon delta) I want to prove that $\lim_{x\to 1}$ $\frac{1}{x-3}=-\frac{1}{2}$
So I started:
$|\frac{1}{x-3} + \frac{1}{2}|$=$\frac{|x-1|}{2|x-3|}$<$\epsilon$
My problem is to bound $\frac{1}{|x-3|}$ from $|x-1|$
Thanks, for any help
| Start by restricting $|x-1| \le 1$. Then $-1 \le x-1 \le 1$ gives
$$
x-3=(x-1)-2 \\
-1-2 \le x-3 \le 1-2 \\
-3 \le x-3 \le -1 \\
|x-3| \ge 1.
$$
Then, whenever $|x-1| \le 1$,
$$
\left|\frac{1}{x-3}+\frac{1}{2}\right|=\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2051854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the probability of drawing the last red ball in the fourth draw? There are 8 Blue(B) and 2 Red(R) balls in a bag. Each time one ball is drawn and replaced by Blue one. What is the probability of drawing the last red ball in the fourth draw ?
I made 3 cases :
A). BBRR = $(\frac{8}{10})^2\cdot \frac{2}{10}\cdot... | Just to generalize a little bit
Assuming $ N $ = Total Number of draws
Notes:
*
*Probability to draw the first red ball is always 2/10
*Probability to draw the second red ball is always 1/10
If $ i \in (1,N-1) $ is the draw you get the first red ball at, then
*
*Probability to draw a blue ball for $ 1 \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2052369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the solution for $x$ with $0 \le x \lt 13$ so $13 \mid 3x^2+9x+7 $. Find the solution for $x$ with $0 \le x \lt 13$ so $13 \mid 3x^2+9x+7 $.
I got this question in my discrete mathematics class.
I don't really get the idea how 13 can divide $3x^2+9x+7 $
All of my friend told me to try all the number between 0 and... | The discriminant of the equation $3x^2+9x+7=0$ equals $81-4 \cdot 9 \cdot 7=-3=10=7^2 \mod 13$. Taking into account that $3^{-1}=9 \mod 13$ we have
$$
x_1=\frac{-9+7}{6}=\frac{-2}{6}=-\frac{-1}{3}=-9=4 \mod 13,
$$
$$
x_2=\frac{-9-7}{6}=\frac{-16}{6}=\frac{-8}{3}=-8 \cdot 9=5 \cdot 9=45=6 \mod 13.
$$
Thus $x=4$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Solution of Logarithmic Inequalities: $\log_{0.5}(\log_{5}(x^2-4))>\log_{0.5}1$
If $\log_{0.5}(\log_5 (x^2-4)) >\log_{0.5}1$ then x lies in the interval:
(a) $(-3,-\sqrt{5})\cup(\sqrt{5}, 3)$
(a) $(-3,-\sqrt{5})\cup(\sqrt{5}, 3\sqrt{5})$
(c) $(\sqrt{5}, 3\sqrt{5})$
(d) $\phi $
I have solved quite a few lo... | First of all, $\log_{0.5}(\log_5 (x^2-4))$ will remain defined iff
$\log_5 (x^2-4)>0\iff x^2-4>5^0=1\iff x^2>5\iff$ either $x>\sqrt5$ or $x<-\sqrt5\ \ \ \ (1)$
$$\log_{0.5}(\log_5(x^2-4))>\log_{0.5}1\iff\dfrac{\log_e(\log(x^2-5))}{\log_e(0.5)}>\dfrac{\log1}{\log_e(0.5)}=0$$
Now as $0.5=\dfrac12=2^{-1},\log_e(0.5)=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2056927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove the inequality $\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$
Prove that the inequality $$\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$$ holds for positive integers $m, n$ if and only if $m$ and $n$ where $m > n$ are two successive terms of the Fibonacci sequence.
I thought about usi... | $\varphi = \frac{1+\sqrt{5}}{2}$ is the root of $x^2-x-1=0$. In fact $x^2-x-1=\left(x - \frac{1+\sqrt{5}}{2}\right) \cdot \left(x - \frac{1-\sqrt{5}}{2}\right)$. Taking $x=\frac{F_{n+1}}{F_n}$, we have:
$$\left|\left(\frac{F_{n+1}}{F_n}\right)^2-\frac{F_{n+1}}{F_n}-1\right|=\left|\frac{F_{n+1}}{F_n} - \frac{1+\sqrt{5}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
what is the value of $ x$? If $\log_{2}3^4\cdot\log_{3}4^5\cdot\log_{4}5^6\cdot....\log_{63} {64}^{65}=x!$, what is the value of $ x$?
I've tried
$$\log_2 3^4\cdot\log_3 4^5\cdot\log_4 5^6\cdot.... \log_{63} 64^{65}$$
$$=\dfrac{4\log 3}{\log 2}\cdot\dfrac{5\log 4}{\log 3}\cdot\dfrac{6\log 5}{\log 4}\cdots\dfrac{65\log ... | Write $\;\;\;\;\;\;\;\dfrac{4\log 3}{\log 2}\cdot\dfrac{5\log 4}{\log 3}\cdot\dfrac{6\log 5}{\log 4}\cdots\dfrac{65\log 64}{\log 63}\;\;\;\;\;$ as
$$=\dfrac{4\log 3}{\log 2}\cdot\dfrac{5\log 4}{\log 3}\cdot\dfrac{6\log 5}{\log 4}\cdots\dfrac{65\cdot 6\log 2}{\log 63}$$
$\{\because \log64=\log2^6=6\log2\}$
Eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2058168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding the linear combination of a vector that is in a span So say we have
Span S =
$
\begin{Bmatrix}
\begin{bmatrix}
1 \\
0 \\
1 \\
\end{bmatrix}
\begin{bmatrix}
-1 \\
1 \\
1 \\
\end{bmatrix}
\begin{bmatrix}
1 \\
... | You can read the solutions in the reduced row echelon form of the augmented matrix:
\begin{align}
\begin{bmatrix}1&-1&1&-1\\0&1&1&4\\1&1&3&7\end{bmatrix}\rightsquigarrow\begin{bmatrix}1&-1&1&-1\\0&1&1&4\\0&2&2&8\end{bmatrix}\rightsquigarrow\begin{bmatrix}1&-1&1&-1\\0&1&1&4\\0&0&0&0\end{bmatrix}\rightsquigarrow\begin{bm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2061691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find the last Digit of $237^{1002}$? I looked at alot of examples online and alot of videos on how to find the last digit But the thing with their videos/examples was that the base wasn't a huge number. What I mean by that is you can actually do the calculations in your head. But let's say we are dealing with a $3$ di... | Finding the last digit of ${237}^{1002}$ is same as finding the last digit of $ 7^{1002}$.
We see that
$7^1 = 7$;
$7^2 = 9$;
$7^3 = 3$;
$7^4 = 1$;
$\dots$
Then comes the repetition of $7$, $9$, $3$ and $1$ as the powers of $7$ increase.
Since our power is even, our choices reduce to last digit of $7^2$ and $7^4$ which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 11,
"answer_id": 9
} |
How many committees of $7$ people can be formed in a class of $14$ boys and $10$ girls if the committee is to contain at least $4$ boys? In a class of $14$ boys and $10$ girls, a committee of $7$ is to be formed. How many committees are possible.
(a) if anybody can serve in the committee?
(b) if the committee is to hav... | Case 1-
C(24,7)
= $\frac{24!}{7!\times17!}$
Case 2-
C(14,4) $\times$ C(10,3)
= $\frac{14!}{4!\times10!} \times \frac{14!}{4!\times10!}$
Case 3-
At least 4 boys, so may be 5 or 6 or 7.
C(14,4) $\times$ C(10,3) + C(14,5) $\times$ C(10,2) + C(14,6) $\times$ C(10,1) + C(14,4)
= $\frac{14!}{4!\times10!} \times \frac{10!}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is there an alternative form/identity for $\prod_{k=1}^{m} \cos(2^kx)$? I am working with a family of integrals that involve a product of cosines of the form $\prod_{k=1}^{m} \cos(2^kx)$. Is there a formula or identity for simplifying $\prod_{k=1}^{m} \cos(2^kx)$ to a polynomial number of terms in $x$? Or $\cos x$, $... | $$\cos x\cdot \cos (2x)\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P$$
Multiply both sides by $\sin x$, so
$$\frac{1}{2}\sin 2x\cdot \cos (2x)\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
$$\frac{1}{2^2}\sin 4x\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
$$\frac{1}{2^3}\sin 8x\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Computing $\lim_{x\to0} \frac 8 {x^8} \left[ 1 - \cos\frac{x^2} 2 - \cos\frac{x^2}4 + \cos\frac{x^2}2\cos\frac{x^2}4 \right]$ without using L'Hospital We have to find the following limit.
$$\lim_{x\to0} \frac 8 {x^8} \left[ 1 - \cos\frac{x^2} 2 - \cos\frac{x^2}4 + \cos\frac{x^2}2\cos\frac{x^2}4 \right]$$
In this I thou... | HINT:
$$1-a-b+ab=(1-a)(1-b)$$
$$\lim_{h\to0}\frac{1-\cos2h}{h^2}=2\lim_{h\to0}\left(\frac{\sin h}h\right)^2 = \text{?}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Do not use series expansion or L' Hospital's rule: $f(x)=\frac{\sin 3x+A\sin 2x+B\sin x}{x^5}$ In the following function
$$f(x)=\frac{\sin 3x+A\sin 2x+B\sin x}{x^5},(x\neq 0)$$
is continuous at $x= 0$. Find $A$ and $B$. Also find $f(0)$.
I first thought of using L hospital. But my sir told me to
'Do not use series exp... | Use $\sin2x=2\sin x\cos x$ and $\sin3x=3\sin x-4\sin^3x$, so your function can be written
$$
\frac{\sin x}{x}\frac{3-4\sin^2x+2A\cos x+B}{x^4}
$$
In order for the limit to be finite, you need that the numerator in the second fraction has limit $0$, so
$$
3+2A+B=0
$$
so $B=-2A-3$. Thus you get, removing $\frac{\sin x}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Partition with minimum size of parts In how many ways can we partition a number $N$? The answer is given by the partition function $p(N)$. So for instance, the partitions of the number $4$ are:
\begin{equation}
4; ~~~ 3+1;~~~ 2+2;~~~ 2+1+1;~~~1+1+1+1
\end{equation}
and hence $p(4) = 5$.
How is this modified if I requi... | Another technique is based upon generating functions. We encode the summands $j$ as powers of $x$ and the coefficient of $x^j$ provides the number of contributions of the summand $j$.
In the example with $N=4$ the number of partitions $p(4)$ is
\begin{align*}
p(4)&=[x^4](x^0+x^1+x^2+x^3+x^4)(x^0+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2069639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Compute $\frac{p(12)+p(-8)}{10}$ where $p(x)=x^4+ax^3+bx^2+cx+d$ I have got an olympiad problem which is as follow:
Compute $\frac{p(12)+p(-8)}{10}$ where $p(x)=x^4+ax^3+bx^2+cx+d$ and $p(1)=10$, $p(2)=20$, $p(3)=30$.
I have been told that answer is $1984$.
I thought applying the values and getting a relation between... | We can write $p(x)-10x=0\forall x=1,2,3$
So using factor theorem $p(x)-10x = 0$ has three factors $(x-1)\;,(x-2)\;,(x-3).$
Given $p(x)$ is a $4^{th}$ degree equation with leading coefficients $=1$
So let $x=r$ be the $4^{th}$ root of above equation
So $$p(x)-10x = (x-1)(x-2)(x-3)(x-r)$$
So $$p(12) = 11\cdot 10\cdot 9\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The sum of all integers from $1$ to $p$ is divisible by $p$ and all prime numbers before $p$. What can $p$ be?
The summation of all integers from $1$ to $p$ is divisible by $p$ and all prime numbers before $p$. Find all possible solutions for $p$, with proof. Here $p$ is a prime number.
This is a preparation probl... | Let $p_n$ be the $n$-th prime.
By Betrand's Postulate, $p_n < 2^n$ for $n > 1$ and therefore $S_n:=\dfrac{p_n(p_n+1)}{2} < 2^{2n-1} + 2^{n-1} < 2^{2n} = 4^n$.
But $5 < p_n$ for $n > 3$ so $P_n:=\displaystyle\prod_{i=1}^n p_i > 5^{n-1}$ for $n > 3$.
Since $P_n$ must divide $S_n$ then $P_n \leq S_n$, so either $5^{n-1} <... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why $x^5y\equiv y^5x\pmod {240}$ if $x,y$ have similar parity? Let $x,y$ be any 2 integers of similar parity. Why do we have :$$x^5y\equiv y^5x\pmod {240}$$
Any hint anyone?
| Factorising we get $xy(x-y)(x+y)(x^2 + y^2) \equiv 0 \mod 2^4\cdot 3\cdot 5$
Because $x, y$ have the same parity then, if $x,y$ even all factors are divisible by 2, else if $x,y$ odd all but $xy$ are divisible by 2 and $(x^2 - y^2)$ is divisible by 8. Therefore $xy(x-y)(x+y)(x^2 + y^2) \equiv 0 \mod 2^4$
Using Fermat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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volume of a truncated cone that is not a frustum I know the formula for a conical frustum is
$$\frac{\pi h}{3}\left( r^2+rR+R^2 \right) $$
What would the formula be for the area of a truncated right circular cone be where the top is not parallel to the base.
With the plane truncating the cone at an angle $\theta$ from... | A plane that meets one nappe of a right circular cone in an ellipse defines an oblique cone with an elliptical base. If the plane lies at distance $d$ from the cone's vertex, and if the base has semi-axes $a$ and $b$, the volume is $\frac{1}{3} \pi abd$.
If the cone has vertex half-angle $\phi$, the cutting plane cross... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the least positive integer that is divisorous?
We say that a positive integer $N$ is divisorous if the ones digits of the positive divisors of $N$ include all of the base-ten digits from $0$ to $9$. What is the least positive integer that is divisorous?
I didn't see an easy way of finding the least such $N$. ... | We first see that such an $N$ must be a multiple of $2$ since otherwise the units digits of the divisors would never be even. We also see that $N$ must be a multiple of $5$ since any number which has a units digit of $5$ must be a multiple of $5$. Thus $N$ is a multiple of $10$ and so we may write $$N = 2 \cdot 5k$$ fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2073950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Proving the inequality I need to prove this:
$\binom{n}{2}a^2 + \binom{m}{2}b^2
\leq
\binom{n+m}{2} \left( \frac{n\cdot a + m\cdot b}{n+m} \right) ^2$
I tried this (WLOG $n\leq m$):
$\binom{n}{2}a^2 + \binom{m}{2}b^2 = \frac{n(n-1)}{2}a^2 + \frac{m(m-1)}{2}b^2
\\
\binom{n+m}{2} \left( \frac{n\cdot a + m\cdot b}{n+m} \... | A full expanding gives $(a-b)^2+2(m+n)ab\geq0$ and the rest for you.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the splitting field of the polynomial $x^5+2x^4+5x^2+x+4$ over $F_{11}$. I am working on the following problem:
Find the splitting field of the polynomial $f(x)=x^5+2x^4+5x^2+x+4$ over $F_{11}$.
What I have done:
So far I found that $-2$ is the only root of $f(x)$ in $F_{11}$. I have also factored $f(x)$ as $f(... | We are considering the field $F_{11}(\alpha,\beta)$ were $\alpha^2 + 5\alpha +1=0$ and $\beta^2 - 5\beta +2=0$. We are going to show that $$F_{11}(\alpha,\beta)=F_{11}(\alpha+\beta)$$
In fact summing the equations we get $$(\alpha + \beta)(\alpha - \beta + 5)=1$$ which means that these elements are inverse one another,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$x + \frac1y = y + \frac1z = z + \frac1x$, then value of $xyz$ is? If $x,y,z$ are distinct positive numbers, such that $$x + \frac1y = y + \frac1z = z + \frac1x $$ then value of $xyz$ is?
$$A)\ 4\quad B)\ 3\quad C)\ 2\quad D)\ 1$$
My attempt:
1.I equaled the equation to '$k$'. Using the AM-GM inequality, I found tha... | By multiplying the first equation by $yz$, the second by $xz$, and the equation connecting the first and third expression by $xy$, we get
$$xyz+z = y^2z+y$$
$$xyz+x = z^2x+z$$
$$xyz+y = x^2y+x$$
Adding these three, cancelling $x+y+z$ from both sides, and dividing by $xyz$ gives $$\frac{x}{z}+\frac{y}{x}+\frac{z}{y} = 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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finding value of $ \int\frac{1}{x^3\sqrt{(1+x)^3}}dx$ finding value of $\displaystyle \int\frac{1}{x^3\sqrt{(1+x)^3}}dx$
Substituting $x=\tan^2 \theta\;, dx = 2\tan \theta \sec^2 \theta$
integral is $=\displaystyle \int \frac{2\tan \theta \sec^2 \theta }{\tan^6 \theta \cdot \sec^3 \theta}d\theta= 2\int\frac{\cos^6 \the... | $$
\left.\begin{aligned}
\int \frac{\cos ^6 \theta}{\sin ^5 \theta}&=-\frac{1}{4} \int \cos ^5 \theta d\left(\frac{1}{\sin ^4 \theta}\right) \\
& =-\frac{\cos ^5 \theta}{4 \sin ^4 \theta}-\frac{5}{4} \int \frac{\cos ^4 \theta \sin ^2 d \theta}{\sin ^4 \theta} \\
& =-\frac{\cos ^5 \theta}{4 \sin ^4 \theta}-\frac{5}{4} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prime factor $>250000$ for $1002004008016032$ I need to find a prime factor, $p$, of $1002004008016032$ such that $p \gt 250000$. Now this is a very large number and I know it would be stupid of me to factorize such a large number into its prime factors . Any help how I can solve this ?
| $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+xy^{n-2}+y^{n-1})$
Taking $x=10^3,y=2$, we get
$1002004008016032=x^5+x^4y+x^3y^2+x^2y^3+xy^4+y^5=\frac{x^6-y^6}{x-y}$
Also,
$\frac{x^6-y^6}{x-y} = (x+y)(x^2+xy+y^2)(x^2-xy+y^2)$
=$1002\times1002004\times998004 = 1002\times16\times250501\times249501$
Hence, the prime n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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prove $\sum_{k=0}^{3n+2}\frac{(-1)^k}{6n+5-k}\binom{6n+5-k}{k}=\frac{1}{6n+5}$ I want to
prove that $$\sum_{k=0}^{3n+2}\frac{(-1)^k}{6n+5-k}\binom{6n+5-k}{k}=\frac{1}{6n+5}$$
Let $\displaystyle f_n(x)=\sum_{k=0}^{\lfloor (n+1)/2\rfloor}\binom{n-k+1}{k}x^{n-k}$,
then we have a recurrence relation
$$f_n(x)=xf_{n-1}(x)+... | Chebyshev polynomials of the second kind have the following closed form:
$$ U_n(x) = \sum_{r=0}^{\lfloor n/2\rfloor}\binom{n-r}{r}(-1)^r (2x)^{n-2r} \tag{1} $$
hence:
$$ x^{n-1}\,U_n(x/2) = \sum_{r=0}^{\lfloor n/2\rfloor}\binom{n-r}{r}(-1)^r (x)^{2n-2r-1} \tag{2} $$
and:
$$ S_n=\sum_{r=0}^{\lfloor n/2\rfloor}\binom{n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integral of $\sqrt{2+t^2}$ How can I solve the integral of
$$ \int \sqrt{2+t^2}\,dt$$
I obtained it by calculating the length of the arc $a(t)=(t\cos t,t \sin t,t)$ with $t \in [0,2\pi]$.
I try with substitution with the function $\sinh$ but i don't manage to resolve it.
| One way is to integrate by parts:
$$
\int_0^{2\pi} \underbrace{\sqrt{t^2 + 2}}_u \, \underbrace{dt}_{dv}
= \underbrace{\sqrt{t^2 + 2}}_u \cdot \underbrace{t}_v \bigg|_0^{2\pi}
- \int_0^{2\pi} \underbrace{t}_v \cdot \underbrace{\frac t{\sqrt{t^2+2}} \, dt}_{du}
$$
Continuing without the labels:
\begin{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate the Volume integral $\iiint\left(x^2+y^2+z^2\right)\mathbb dv$. Suppose $a>0$ and $S = \{(x,y,z) \in \mathbb R^3 : x^2+y^2+z^2=a^2\}$ then
MY FIRST APPROACH: $$\int\int\int\left(x^2+y^2+z^2\right)\mathbb dv=a^2\int\int\int \mathbb dv=a^2.\frac{4}{3}\pi a^3=\frac 4 3\pi a^5$$.
MY SECOND APPROACH:If i use spher... | Note
$$S = \left\{(x,y,z) \in \mathbb R^3 : x^2+y^2+z^2\le a^2\right\}=\left\{(\theta,\phi,\rho) : 0\le \theta\le 2\pi\, ,\,0\le \phi\le\pi\,\,, \rho^2=a^2\right\}$$
set
\begin{cases}
x=\rho\sin\phi\cos \theta\\
y=\rho\sin\phi\sin \theta\\
z=\rho\cos\phi
\end{cases}
we have
$$\left|\frac{\partial(x,y,z)}{\partial(\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that given fraction is power of two I do not really know where to start with the following:
Prove that
$$\frac{\displaystyle{ \prod_{k = 1}^{2n - 1} k^{\min(k, 2n - k)}}}{\displaystyle{\prod_{k = 1}^{n - 1}(2k + 1)^{2n - 2k - 1}}}$$ is a power of two.
Could you give me a hint which helps to solve this pr... | Much calculus there :
$$\begin{aligned}\frac{\displaystyle{ \prod_{k = 1}^{2n - 1} k^{\min(k, 2n - k)}}}{\displaystyle{\prod_{k = 1}^{n - 1}(2k + 1)^{2n - 2k - 1}}}
&= \frac{\displaystyle{ \prod_{k = 1}^{n} k^{\min(k, 2n - k)}} \times \prod_{k = n + 1}^{2n - 1} k^{\min(k, 2n - k)}}{\displaystyle{\prod_{k = 1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Additivity of $\arctan(\frac{x+y}{1-xy})$ Show in two different way that,
$$\arctan\left({\frac{x+y}{1-xy}}\right) = \arctan\left(x\right) + \arctan\left(y\right)
$$
The first way I know derive from the addition formula of tangent
$$ \frac{ \tan\left(x\right)+ \tan\left(y\right)}{1- \tan(x) \tan(y)} = \tan\left(x+y \r... | Differentiating,
$$ \begin{align}
\frac{d}{dx} \arctan{\left( \frac{x+y}{1-xy} \right)} &= \frac{1}{1+(x+y)^2/(1-xy)^2} \left( \frac{1}{1-xy} - \frac{-(x+y)y}{(1-xy)^2} \right) \\
&= \frac{ 1-xy + xy+y^2 }{(1-xy)^2 + (x+y)^2} \\
&= \frac{ 1+y^2 }{1-2xy+x^2y^2+x^2+2xy+y^2} \\
&= \frac{ 1+y^2 }{1+x^2+y^2+x^2 y^2} \\
&= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Verify $\cos{x}<\left(\frac{\sin{x}}{x}\right)^3$ for $0Verify by Mean Value Theory or otherwise that $\cos{x}<\left(\frac{\sin{x}}{x}\right)^3$ for $0<x<\pi/2$.
I am unable to solve the problem. Please give me a solution of the problem.
| For $x > 0$ the well-known representation as alternating series
give the estimates
$$
\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} > x - \frac{x^3}{6} \, ,\\
\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} < 1 - \frac{x^2}{2} + \frac{x^4}{24} \, .
$$
For $0 < x < \pi/2$ in particular $x^2 < 6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Incorrect solution in limits $$
\lim_{n\to \infty}\left(\frac{1}{n^4}{+3^{\frac{2}{2+n}}}\right)^{n}
$$
So i re-write it like:
$\lim_{n\to \infty}e^{n\ln{\frac{1}{n^4}\ln3^{\frac{2}{2+n}}}}$ $=$ $e^{\frac{2n}{2+n}\ln{\frac{1}{n^4}\ln3}}=e^{{2}\ln{\frac{1}{n^4}\ln3}}$
So here, $\ln{\frac{1}{n^4}}$ give us minus infini... | Put $n=\frac{1}{y}$ .The limit reduces to
$L=\lim_{y\to 0}\left(y^4+3^{\frac{2y}{2y+1}}\right)^{\frac{1}{y}}$
Take log on both sides
$\log_{e}L=\lim_{y\to 0} \log_{e}\left(y^4+3^{\frac{2y}{2y+1}}\right)^{\frac{1}{y}}$
$\log_{e}L=\lim_{y\to 0}\frac{1}{y} \log_{e}\left(y^4+3^{\frac{2y}{2y+1}}\right)$
$\log_{e}L=\lim_{y\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2090387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Find the limit of $6^n(2-x_n)$ where $x_n=\sqrt[3]{6+\sqrt[3]{6+\dots+\sqrt[3]{6}}}$ with $n$ roots
Let $x_n=\sqrt[3]{6+\sqrt[3]{6+\dots+\sqrt[3]{6}}}$ where the expression in the RHS has $n$ roots.
Find the following limit: $\lim \limits_{n\to \infty}6^n(2-x_n)$
My approach: I had two approaches. The first one was t... | Lets look at the sequence
$$a_n=6^n(8-x_n^3)=6^n(2-x_n)(4+2x_n+x_n^2)$$
Now $$\lim_{n\to\infty}6^n(2-x_n)(4+2x_n+x_n^2)=\lim_{n\to\infty}12\cdot 6^n(2-x_n)$$ so lets instead look at the $\lim a_n$ and whatever the limit is just multiply it by $12$.
We have that
$$\frac{a_{n+1}}{a_n}=\frac{6^{n+1}(8-x_{n+1}^2)}{6^n(8-x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Proof convergence of $\sum\limits_{n = 1}^\infty\frac {n!} {n^n}$ To prove the convergence of $\sum\limits_{n = 1}^\infty\frac {n!} {n^n}$ I used that $\lim\limits_{n\to\infty}|\frac {a_{n+1}} {a_n}|$ has to be $<1$:
$$\lim\limits_{n\to\infty}|\frac {a_{n+1}} {a_n}|$$
$$=\lim\limits_{n\to\infty}|\frac {\frac {(n+1)!} {... | We can also use the Comparison Test: $$0\le \frac{n!}{n^n}=\frac{1\cdot 2\cdot 3\cdot\ldots\cdot n}{n\cdot n\cdot n\cdot\ldots\cdot n}\le \frac{1\cdot 2\cdot n\cdot\ldots \cdot n}{n\cdot n\cdot n\cdot\ldots\cdot n}=\frac{2}{n^2},$$ and $\displaystyle \sum_{n=1}^{+\infty}\frac{2}{n^2}$is convergent, so the given series ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Proving a sequence is convergent by convergence of odd and even subsequences? I want to use this method to prove convergence of: $a_{n}=\sqrt{2-a_{n-1}}$, $a_{0}=\frac{2}{3}$.
Here is my attempt at proof:
It can be proven inductively that $0< a_{n}<2$ for all $n$.
I want to show that each of the odd and even subsequenc... | If $0<x<1$ then $\sqrt {1-x}>1-x/(2-x)$ (by squaring both sides and calculating). This deserves more publicity. We will use this in (2).(ii). below.
If $a_n$ converges to $L$ then $0\leq L=\sqrt {2-L}\implies (L^2=2-L\land 0\leq L)\implies L=1.$ So it is natural to let $a_n=1+b_n.$
We have $0<|b_n|<1.$ (Equivalent to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2094713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
How to calculate approximation of integral by developing integrant into power series? How to calculate approximation $I = \int_{-1}^{1} e^{-x^{2}} $ by developing integrant into power series, and how many terms of series is needed, so error is smaller than $\varepsilon = 10^{-3}$ ?
We can rewrite this $I = \int_{-1}^{... | Since
$$
e^{-x^2}=\sum_{n=0}^\infty\dfrac{(-x^2)^n}{n!}=\sum_{n=0}^\infty\dfrac{(-1)^n}{n!}x^{2n},
$$
we have
$$
I=\int_{-1}^1e^{-x^2}\,dx=2\int_0^1e^{-x^2}\,dx=2\sum_{n=0}^\infty\frac{(-1)^n}{n!(2n+1)}\equiv \sum_{n=0}^\infty(-1)^nb_n
$$
with
$$
b_n=\dfrac{2}{n!(2n+1)}.
$$
Since the sequence $\{b_n\}$ is positive and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2095143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Easy way to decompose $\frac{1}{X^{3}\cdot (X-2)^{3}}$ into partial fractions? Is there any easy way or shortcut to decompose $$\frac{1}{X^{3}\cdot (X-2)^{3}}$$ into partial fractions ? because dealing with the usual way of replacing and giving values to $X$ is too clumsy in this case , so I was wondering if there ... | We may consider that:
$$ \frac{1}{t-1}-\frac{1}{t+1} = \frac{2}{(t-1)(t+1)}\tag{1} $$
so:
$$\begin{eqnarray*} \frac{8}{(t-1)^3 (t+1)^3} &=& \frac{1}{(t-1)^3}-\frac{3}{(t-1)^2(t+1)}+\frac{3}{(t-1)(t+1)^2}-\frac{1}{(t+1)^3}\\&=&\frac{1}{(t-1)^3}-\frac{1}{(t+1)^3}+\frac{3}{2}\frac{2}{(t-1)(t+1)}\left(\frac{1}{t+1}-\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2095294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Alternative form of Eisenstein integers I just recently got into number theory and algebra so my knowledge is very limited. I stumbled upon the Eisenstein integers $\mathbb{Z}[\omega]$. They are of the form $\varepsilon = a + b \omega$, where $\omega = e^{2 \pi i/3} = -1/2 + i \sqrt{3}/2$.
Now my problem:
I want to sh... | I think this excessive emphasis on $\omega$ often serves to confuse students. Don't get me wrong, $\omega$ is important. $$\omega = -\frac{1}{2} + \frac{\sqrt{-3}}{2},$$ and $N(\omega) = 1$, which means that $\omega$ is a unit. And furthermore $\omega$ is a complex cubic root of 1.
Let's review the norm function for nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2097658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Proof of an inequality using Cauchy's inequality: $\sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2\geq n^3+2n+\frac{1}{n}$ Given that $p_k> 0$ and $p_1+p_2+\cdots+p_n=1$, prove that
\begin{equation}
\sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2\geq
n^3+2n+\frac{1}{n}.
\end{equation}
I believe that Cauchy's inequality sho... | The question is from chapter 1 of J. Michael Steele's book ``The Cauchy-Schwarz Master Class''. I didn't know that the book has solutions at the back. Below is the author's solution, which only employs Cauchy's inequality.
First, we expand the square.
\begin{equation}
\sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2 =
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2101222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Finding $\lim_{x\rightarrow 0}\lfloor \frac{2017 \sin x}{x} \rfloor +\lfloor \frac{2017 \tan x}{x} \rfloor $ Finding value of $\displaystyle \lim_{x\rightarrow 0}\bigg\lfloor \frac{2017 \sin x}{x}\bigg \rfloor +\bigg\lfloor \frac{2017 \tan x}{x}\bigg \rfloor,$ where $\lfloor x \rfloor $ is floor function of $x$
Attempt... | Yes, the key is that $\sin x<x<\tan x$ for $x\to0^{+}$, which implies $\displaystyle \frac{\sin x}{x}<1<\frac{\tan x}{x}$. Since after division both expressions are even functions, this double inequality is true both as $x\to0^{+}$ and as $x\to0^{-}$. We also know that both $\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Intersection of 3 circles on equilateral triangle given the difference in their radii Imagine we are given three intersecting circles centered on the vertices of an equilateral triangle of side length $1$, with $(0,0)$ arbitrarily placed at the bottom left corner. The circles have radii $r$, $r+a$, and $r+b$ respectiv... | Consider a general point $(x,y)$ inside the triangle,
and let the distance of $(x,y)$ from the vertices $(0,0),$
$\left(\frac12,\frac{\sqrt3}2\right),$ and $(1,0)$ be
$r_1$, $r_2,$ and $r_3$ respectively.
(That is, each of $r_1$, $r_2,$ and $r_3$ is a function of $(x,y)$.)
We want to find a point $(x,y)$ such that
$r_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2103984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
If $f(g(x)) = 4x^2-8x$ and $f(x)=x^2-4$, then what's the value of $g(x)$? I'm a little stuck with this simple function:
If ${f(g(x)) = 4x^2-8x}$ and ${f(x)=x^2-4}$, then what's the value of ${g(x)?}$
Any tips?
| \begin{align}
f(g(x))&=4x^2-8x\\
g(x)^2-4&=4x^2-8x\\
g(x)^2&=4x^2-8x+4\\
&=4(x^2-2x+1)\\
&=4(x-1)^2\\
&=[\ \!2(x-1)\ \!]^2\\
\implies g(x)&=\pm2(x-1)
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2104899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Infinite sum including logarithm I would like to calculate the following sum:
$$\sum_{n=1}^{\infty}\ln \left( \frac{n^2+2n+1}{n^2+2n} \right)$$
I do know that it converges but I have gone that far:
\begin{align}
& \sum_{n=1}^{\infty}\ln \left( \frac{n^2+2n+1}{n^2+2n} \right) \Longleftrightarrow \sum_{n=1}^\infty \ln ... | $\begin{array}\\
\sum_{n=1}^{m}\ln \left( \frac{n^2+2n+1}{n^2+2n} \right)
&=\sum_{n=1}^{m}\ln \left( \frac{(n+1)^2}{n(n+2)} \right)\\
&=\sum_{n=1}^{m}( (2\ln(n+1)-\ln(n)-\ln(n+2))\\
&=2\sum_{n=1}^{m}\ln(n+1)-\sum_{n=1}^{m}\ln(n)-\sum_{n=1}^{m}\ln(n+2)\\
&=2\sum_{n=2}^{m+1}\ln(n)-\sum_{n=1}^{m}\ln(n)-\sum_{n=3}^{m+2}\ln... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2105352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.