Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Divisibility of integers Let $n > 1$ be an integer. Then $2^n - 1\nmid 3^n - 1$. I don't know how to prove it. Can anybody help me, please?
In general, for a fixed positive integer $a > 1$, has $a^n - 1|(a +1)^n - 1$ any integer solutions?
| The idea used below is very close to the one used by @Zander. It will not be a surprise to those who have seen my other posts that the details take longer.
If $n$ is even, then $2^n-1$ is divisible by $3$, so $2^n-1$ cannot divide $3^n-1$ unless $n=0$.
So let $n>1$ be odd. Let $p$ be a prime that divides $2^n-1$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Absolute maximum value of $\sin^2(x)-\sin(x)$ in $[0,\frac{3\pi}{2}]$ I thought it does not have absolute maximum, but wanted to just check and see why
| A continuous function on a closed interval always has both an absolute max and an absolute min. And, the absolute max and min can only occur at critical points or the end points of the interval (I know of at least one book that includes end points as critical points). Therefore, the strategy that calculus books give ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How I can find the value of $abc$ using the given equations? If I have been given the value of
$$\begin{align*}
a+b+c&= 1\\
a^2+b^2+c^2&=9\\
a^3+b^3+c^3 &= 1
\end{align*}$$
Using this I can get the value of
$$ab+bc+ca$$
How i can find the value of $abc$ using the given equations?
I just need a hint.
I have tried ... | First $\displaystyle{(a+b+c)^2 = (a^2+b^2+c^2)+2(ab+bc+ca)}$, which implies $1 = 9+2X$ where $X=(ab+bc+ca) \implies (ab+bc+ca)=-4$
Using $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$, and substituting the known values $1-3Y = 9+4$ and solve for $Y=abc$
Note: You can always check with Wolfram Alpha if your answer is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
The solution set of the equation $|2x - 3| = - (2x - 3)$ The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is
$A)$ {$0$ , $\frac{3}{2}$}
$B)$ The empty set
$C)$ (-$\infty$ , $\frac{3}{2}$]
$D)$ [$\frac{3}{2}$, $\infty$ )
$E)$ All real numbers
The correct answer is $C$
my solution:
$\ 2x-3 = -(2x-3)$... | *
*case - when $|2x-3| = -2x+3$ so than there will be $-2x+3=-2x+3$ --- from what will result $0=0$ so for this case the answer will be E).
2.case - when $|2x-3| = 2x-3$ so than there will be $2x-3=-2x+3$ --- 4x=6 --- $x=6/4 =3/2$ --- so in this case $x=3/2$
Note that because in your exercise , in this equation the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Express some equations as polynomial equations Given
$$\begin{align*}
x&=(2+\cos(2s))\cos(3s)\\
y&=(2+\cos(2s))\sin(3s)\\
z&=\sin(2s),\end{align*}$$
I was wondering how to express these equations as polynomial equations in $x$, $y$, $z$, $a=\cos(s)$, $b=\sin(s)$.
Thanks!
Edit: I expect that the polynomial equations... | $$\begin{align*}
x^2&=(2+\cos(2s)^2\cos(3s)^2\\
y^2&=(2+\cos(2s)^2\sin(3s)^2\\
x^2+y^2&=(2+\cos(2s))^2(\cos(3s)^2+sin(3s)^2)=(2+\cos(2s))^2\\
x^2+y^2&=(2+\cos(2s))^2=4+4\cos(2s)+\cos(2s)^2=4+4\cos(2s)+(1-\sin(2s)^2)\\
x^2+y^2&=5+4\cos(2s)-z^2\\
x^2+y^2+z^2&=5+4(\cos(s)^2-\sin(s)^2)\\
x^2+y^2+z^2&=5+4a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Progessive terms adding and minusing What is the value of
$$1+2+3-4+5+6+7-8+9+10+11+12...+97+98+99-100 \ ?$$
Any help is appreciated, thank you!
I added the terms as an AP then subtracted 10 then all the numbers that were missed out, not sure if this is right though.
| If you rearrange the terms, you can write it as:
$1+2+3+5+6+7+9+10+...+98+99-4-8-...-100=1+2+3+4+...+100-2(4+8+12+...+100)=
1+2+...+100-2\cdot 4(1+2+...+25)=\frac{100(100+1)}{2}-2\cdot4\frac{25(25+1)}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/123003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
limit of $\lim_{x \to 0}\left ( \frac{1}{x^{2}}-\cot x\right )$ Help me with that problem, please.
$$\lim_{x \to 0}\left ( \frac{1}{x^{2}}-\cot x\right )$$
| Note that for $x>0$ near $0$ we have
$$
\frac{1}{x^2} - \cot x = \frac{1}{x^2}-\frac{\cos x}{\sin x} \geq \frac{1}{x^2}-\frac{1}{\sin x} = \frac{\sin x - x^2}{x^2 \sin x}.
$$
Then we have
$$
\lim_{x\to 0^+}\frac{\sin x - x^2}{x^2 \sin x}
\ \operatorname*{=}^{\small\mathrm{L'H}}\ \lim_{x\to 0^+} \frac{\cos x-2x}{2x\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/123324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Factorial Divisors Q. For what maximum value of $n$ will the expression $\frac{10200!}{504^n}$ be an integer? I have the solution to this question and I would like you to please go through the solution below. My doubt follows the solution :)
The solution can be found by writing $504 = 2^3 \cdot 3^2 \cdot 7$ and then f... | In this answer, it is shown that the number of factors of $p$ in $n!$ is
$$
\frac{n-\sigma_p(n)}{p-1}\tag{1}
$$
where $\sigma_p(n)$ is the sum of the base-$p$ digits of $n$.
Factor
$$
504=2^3\cdot3^2\cdot7\tag{2}
$$
Write $10200$ in base-$2$, base-$3$, and base-$7$:
$$
\begin{array}{}10011111011000_2&111222210_3&41511_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find the value of : $\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$
Possible Duplicate:
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
$$\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$$
So, I have an intermediate form of $\infty - \infty$ and I tried multipl... | $$\begin{align}
\sqrt{4x^2 + 4} - (2x+2) & = \frac{(\sqrt{4x^2 + 4} - (2x+2))(\sqrt{4x^2 + 4} + (2x+2))}{\sqrt{4x^2 + 4} + (2x+2)}\\
& = \frac{4x^2 +4 - 4(x+1)^2}{\sqrt{4x^2 + 4} + (2x+2)} = - \frac{8x}{\sqrt{4x^2 + 4} + (2x+2)}\\
& = - \frac{8}{\sqrt{4 + 4/x^2} + 2 + 2/x}
\end{align}
$$
Now take the limit as $x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Rearranging a formula, transpose for A2 - I'm lost Given the formula:
$$ q = A_1\sqrt\frac{2gh}{(\frac{A_1}{A_2})^2-1} $$
Transpose for $A_2$
I have tried this problem four times and got a different answer every time, none of which are the answer provided in the book. I would very much appreciate if someone could show ... | $$
\begin{eqnarray*}
q &=& A_1\sqrt{\frac{2gh}{(\frac{A_1}{A_2})^2-1} }&\biggr| : A_1, (\;\;)^2\\
\left(\frac{q}{A_1}\right)^2 &=& \frac{2gh}{(\frac{A_1}{A_2})^2-1}&\biggr| (\;\;)^{-1},\cdot 2gh,+1 \\
2gh\left(\frac{A_1}{q}\right)^2+1 &=& (\frac{A_1}{A_2})^2&\biggr| (\;\;)^{-1/2},\cdot A_1\\
\pm\frac{A_1}{\sqrt{2g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Proving that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$ Let $a$ and $b$ be positive reals. Show that
$$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
| An elementary proof.
We use the Taylor series $e^x = 1 + x + O(x^2)$ and the fact that $\lim_{n\to\infty}(1+x/n)^n = e^x$.
If $a=b$ the identity is trivial.
Without loss of generality, assume $0<a<b$.
Then
$$\begin{eqnarray*}
\left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n
&=& b \left(\frac{1+(\frac{a}{b})^{1/n}}{2}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 9,
"answer_id": 0
} |
How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?. How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ without using a calculator.
Related question: how do we prove that $\cos(\pi/5)\cos(2\pi/5) = 0.25$, also without using a calculator
| I was not able (yet) to follow Chandrasekar's solution, but noticed this while trying to understand the argument (how it could possibly lead to the solution, or how exactly he arrives at $2x^2-x-1$ for $x=\cos\frac{\pi}{5}$, which to me seems non-obvious and even a fallacious deduction from his equations and prose -- a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 18,
"answer_id": 2
} |
How to transform the factored form of $\sin(x)$? We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$.
So $\sin(x)$, if interpreted as a polynomial, could be written as:
$a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too:
$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$
So, the questio... | The proposal is:
$$\sin x = a(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)(x-3\pi)(x+3\pi)\cdots$$
The standard result, already posted by Sam, is in effect
$$
\sin x = \frac{x-\pi}{\pi}\cdot \frac{x+\pi}{\pi} \cdot \frac{x - 2\pi}{2\pi}\cdot \frac{x+2\pi}{2\pi}\cdot\frac{x-3\pi}{3\pi}\cdot\frac{x+3\pi}{3\pi} \cdots
$$
So the coeffici... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Finding the linear combinations of two vectors I am studying for my finals and I'm trying to answer the following question:
Consider the following two vectors in $\mathbb{R}^3$: $a=(1,2,3)$ and $b=(2,3,1)$. Decide
whether it is possible to express the vector $c=(2,4,5)$ as a linear
combination of $a$ and $b$.
I ... | $c=m\cdot a+n \cdot b$
Hence :
$(2,4,5)=m\cdot(1,2,3)+n\cdot(2,3,1)$
So we have following system of equations :
$\begin{cases}
m+2n=2 \\
2m+3n=4 \\
3m+n=5
\end{cases}$
which has no solution ,therefore you cannot express $c$ as linear combination of $a$ and $b$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/134360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$ $$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$$
Tried substitution ($u = \cos\frac{x}{2}$), but I get $-\frac{\cos^3\frac{x}{2}}{3}$ ($-\frac{2}{3}$) instead of the correct answer, which is $1\frac{1}{3}$
| You can successively reduce this to simple integrals using sum formulae,
$$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2} dx = \frac{1}{2}\int_0^{2\pi}\sin(x)\cos\frac{x}{2} dx = \frac{2}{4}\int_0^{\pi}\left(\sin\frac{3x}{2} + \sin\frac{x}{2}\right) dx$$
This will give you, $\frac{1}{2} \times \frac{8}{3} = \frac{4}{3}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why is this equation correct? There is this equation:
$$M = \sum\limits_{i = 1}^{\log n} {\frac{{in}}{{{2^i}}}} = n\sum\limits_{i = 1}^{\log n} {\frac{i}{{{2^i}}}} \leqslant n\sum\limits_{i = 1}^\infty {\frac{i}{{{2^i}}} = 2n} $$
And I don't understand why the rightmost summation can be simplified to 2n. Can you pl... | Knowing that:
$$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = 1 $$
And hence by removing successive terms from the left:
$$ \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \frac{1}{2} $$
$$ \frac{1}{8} + \frac{1}{16} + \cdots = \frac{1}{4} $$
$$ \vdots $$
We have:
$$\begin{eqnarray*}
\sum_{i=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to find $\int \sqrt{a^2-x^2} dx$, where $a$ is a constant How to find $$\int \sqrt{a^2-x^2} dx\;,$$ where $a$ is a constant?
It appears to be $$\frac{\pi a}{2}\;,$$ but how do I get there?
| One way of finding the definite integral is to consider the shape of the graph. If $y = \sqrt{a^2-x^2}$ then $y^2=a^2-x^2$, so $x^2+y^2=a^2$. Apply the Pythagorean theorem: that's the equation of a circle of radius $a$. $y = \pm\sqrt{a^2-x^2}$ is the whole circle; $y = \sqrt{a^2-x^2}$ is the top half of the circle. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
Evaluate the integral: $$ \int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx$$
The answer is $0,$ but I am unable to get it. There is some symmetry I can not see.
| Let $m, n > 0$. Then observe that
$$ \int_{0}^{1} \sqrt[n]{1-x^m} \; dx$$
is the area of the region given by inequalities
$$ 0 \leq x \leq 1 \quad \text{and} \quad 0 \leq y \leq \sqrt[n]{1-x^m}.$$
But the last inequality is equivalent to $0 \leq x^m + y^n \leq 1$. Thus
$$ \int_{0}^{1} \sqrt[n]{1-x^m} \; dx = [\text{Are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/139393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 3,
"answer_id": 0
} |
Represent lengths rectangle using given terms In a rectangle, $GHIJ$, where $E$ is on $GH$ and $F$ is on $JI$ in such a way that $GEIF$ form a rhombus. Determine the following: $1)$ $x=FI$ in terms of $a=GH$ and $b=HI$ and $2)$calculate $y=EF$ in terms of $a$ and $b$.
| We have $GHIJ$ is rectangle and $GEIF$ is a rhombus and also $GH = a = IJ$ and $HI = b=GJ$
We have to find $x=FI$ (side of rhombus) and $y=EF$ (one of the diagonals of rhombus)
(Here I have to draw picture of your problem. I know diagram. But i am not able to draw a picture in mac OS X. You can draw diagram easily)
$$x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/144845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding the integral of $(1+2^{2x})/2^x$
Evaluate the integral:
$$\int\frac{1+2^{2x}}{2^x}\,dx = \int \frac{ 1 + (2^x)^2}{2^x}\,dx$$
Let $u = 2^x$. Then $du = 2^x\ln2\,dx$, which yields $\frac{du}{2^x\ln2} = dx$ so
$$ \int \frac{ 1 + (2^x)^2}{2^x}\,dx = \int \frac{1+u^2}{u}du =
\left( x+ \frac{u^3}{3} \right)\ln... | Your change of variable is fine; your substitution is not quite right and your integral is not quite right. If $u=2^x$, then $du = 2^x\ln(2)\,dx = u\ln(2)\,dx$, so $dx = \frac{du}{\ln(2)u}$. So we have:
$$\begin{align*}
\int \frac{1+2^{2x}}{2^x}\,dx &= \int\frac{1 + (2^x)^2}{2^x}\,dx\\
&= \int\left(\frac{1 + u^2}{u}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to evaluate $\int\limits^1_0 \sqrt{1+\frac{1}{x}}\, \text{d}x$ I need to calculate the length of a curve $y=2\sqrt{x}$ from $x=0$ to $x=1$.
So I started by taking $\int\limits^1_0 \sqrt{1+\frac{1}{x}}\, \text{d}x$, and then doing substitution: $\left[u = 1+\frac{1}{x}, \text{d}u = \frac{-1}{x^2}\text{d}x \Rightarro... | Put $x= \tan^{2}\theta$, then you have the integral as
\begin{align*}
\int_{0}^{1} \sqrt{1+\frac{1}{x}} \ dx &= \int_{0}^{\pi/4} \sqrt{\frac{1+\tan^{2}\theta}{\tan^{2}\theta}} \cdot 2\tan\theta \cdot\sec^{2}\theta \ d\theta \\\ &= 2 \cdot\int_{0}^{\pi/4} \sec^{3}\theta \ d\theta
\end{align*}
Now integrate this funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 2
} |
Summing $\sum_{n=1}^{\infty} \sin\frac{n!\pi}{120}$ How do I sum $$\sum_{n=1}^{\infty} \sin\frac{n!\pi}{120}$$
| Note that $\sin \left(\dfrac{n! \pi}{120} \right) = 0$ for all $n \geq 5$. Hence, $$\sum_{n=1}^{\infty} \sin \left(\dfrac{n! \pi}{120} \right) = \sin \left(\dfrac{1! \pi}{120} \right) + \sin \left(\dfrac{2! \pi}{120} \right) + \sin \left(\dfrac{3! \pi}{120} \right) + \sin \left(\dfrac{4! \pi}{120} \right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/152357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How can we produce another geek clock with a different pair of numbers? So I found this geek clock and I think that it's pretty cool.
I'm just wondering if it is possible to achieve the same but with another number.
So here is the problem:
We want to find a number $n \in \mathbb{Z}$ that will be used exactly $k \in \... | Now with $n = 5$ and $k = 5$.
With $n = 5$ and $k = 5$ (missing a $9$ for now but I'll come back to it later).
$\dfrac{55}{5}-5-5=1$
$\dfrac{5+5}{5}-5+5=2$
$\dfrac{5+5}{5}+\frac{5}{5}=3$
$\dfrac{5+5+5+5}{5}=4$
$5 - 5 + 5 - 5 + 5 = 5$
$5 + \dfrac{5}{5} - 5 + 5 = 6$
$5 + \dfrac{5}{5}+\dfrac{5}{5} = 7$
$5 + 5 - \dfrac{5+5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 10,
"answer_id": 3
} |
Summation of $ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$ till $n$ terms What is the pattern in the following?
*
*Sum to $n$ terms of the series: $$ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$$
| So, what you want is basically $$\left(1 - \frac {1} {2}\right) + \left(1 - \frac {1} {4}\right) + \cdots + \left(1 - \frac {1} {2^n}\right) = n - \left(\frac {1} {2} + \frac {1} {4} + \cdots + \frac {1} {2^n}\right) = n - 1 + \frac {1} {2^n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/153302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Integral of $\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$ I am trying to find $$\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$$
$t = \sec \theta$ $dt = \sec \theta \tan\theta $
$$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \sqrt{\sec^2 \theta-1}}$$
$$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \tan^2 \theta}$$
$$\int_\sqrt{2}^... | $$\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$$
Substitute :
$t = \sec \theta$
$dt = \sec \theta \tan\theta d\theta$
The limits will also change accordingly
When $t=2$ , $\theta = \ arc sec(2) = \frac{\pi}{3}$
When $t=\sqrt2$ , $\theta = \ arc sec(\sqrt2) = \frac{\pi}{4}$
$$=\int_\frac{\pi}{4}^\frac{\pi}{3} \frac{\sec ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Evaluating $\int_{0}^{1} \frac{x^{2} + 1}{x^{4} + 1 } \ dx$ How do I evaluate $$\int_{0}^{1} \frac{x^{2} + 1}{x^{4} + 1 } \ dx$$
I tried using substitution but I am getting stuck. If there was $x^3$ term in the numerator, then this would have been easy, but this one doesn't.
| Another way, if you want to sweat harder instead of the elegant suggestion of Chandrasekhar:$$x^4+1=(x^2+\sqrt{2}\,x+1)(x^2-\sqrt{2}\,x+1)\Longrightarrow$$$$ \frac{x^2+1}{x^4+1}=1-\frac{\sqrt 2\,x}{x^2+\sqrt 2\,x+1}+1+\frac{\sqrt 2\,x}{x^2-\sqrt 2\,x+1}$$ so for example$$\int\frac{\sqrt 2\,x}{x^2+\sqrt 2\,x+1}dx=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
| Let us consider $$A=\iint_{[0,1]^2}\frac{x}{(1+xy)(1+x^2)}dx dy$$
By Fubini's theorem, we have : $$A=\int_0^1\left[\frac{1}{1+x^2}\int_0^1\frac{x\,dy}{1+xy}\right]dx=\int_0^1\frac{\ln(1+x)}{1+x^2}dx$$ and$$A=\int_0^1\left[\int_0^1\frac{x}{(1+xy)(1+x^2)}dx\right]dy$$But$$\frac{x}{(1+xy)(1+x^2)}=\frac{1}{1+y^2}\left(\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 8,
"answer_id": 5
} |
Evaluate the integral $H(y)=\int_{z=1}^{\infty} \frac{1}{z^4+zy}\,dz$
*
*$y\geq0$ define $$H(y)=\int_{z=1}^{\infty} \frac{1}{z^4+zy}\,dz$$ Show that $H$ is a continuous function of $y$ and show $\lim\limits_{y \to +\infty}H(y)=0$.
| This is kind of a brute force method where we explicitly find the function $f(a)$.
$$f(a) = \begin{cases} \dfrac{\log(1+a)}{3a} & \text{if }a >0\\ \dfrac13 & \text{if }a=0 \end{cases}$$
This can be obtained as shown below. We have that for $a>0$, $$\dfrac1{x^4+ax} = \dfrac1{x(x^3+a)} = \dfrac1{ax} - \dfrac{x^2}{a(a+x^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work:
$\displaystyle\left(1 + \exp\frac{2i\pi}{5... | $$
\color{red}{\mathbf 5}\cdot\sum_{k\geqslant0}{n\choose \color{red}{\mathbf 5}k}=\sum_{\ell=1}^{\color{red}{\mathbf 5}}\left(1+\mathrm e^{2\mathrm i\pi\ell/\color{red}{\mathbf 5}}\right)^n=\sum_{\zeta\in\mathbb C\,:\,\zeta^\color{red}{\mathbf 5}=1}\left(1+\zeta\right)^n
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
What other substitutions could I use to evaluate this integral? Consider the integral
$$ \int x^2\sqrt{2 + x} \, dx$$
I need to find the value of this integral, yet all its (seemingly) possible substitutions don't allow me to cancel appropriate terms. Here are three substitutions and their outcomes, all of which cover ... | Your first substitution should work. That is let $u=2+x$, then $ x= u-2$ and thus $$x^2 = (u-2)^2.$$ You should get $$ \int \left(u^{5/2}-4u^{3/2}+4u^{1/2}\right)~du. $$
The second substitution also works. Let $u =\sqrt{2+x}$. So $u^2 = 2+x$. Then $x^2 = (u^2-2)^2$. Putting everything together gives you $$\int 2\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$ I need to prove that for any real number $a>1$ and $b>1$ the following inequality is true:
$$\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$$
| $AM-GM$:
$\frac{a^2}{b-1}+4(b-1)\geq 4a$
$\frac{b^2}{a-1}+4(a-1)\geq 4b$
$\Rightarrow \frac{a^2}{b-1}+4(b-1)+\frac{b^2}{a-1}+4(a-1)\geq 4a+4b$
$\Rightarrow \frac{a^2}{b-1}+\frac{b^2}{a-1}\geq 8$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 2
} |
Under what situations does $x+1$ divide $4n^2-x$? I am looking at the equation
$$\frac{4n^2-x}{x+1} = y$$
for even $x$ and $y$, both positive. Under what situations does $x+1$ divide $4n^2-x$?
| I will throw another wooden nickel on the fire.
Since $\dfrac{4n^2-x}{x+1}+1=\dfrac{4n^2+1}{x+1}$, we get that the question is equivalent to asking when
$$
x+1\,|\,4n^2+1\tag{1}
$$
Trivially, $x=0$ and $x=4n^2$ satisfy $(1)$, and these are the only solutions when $4n^2+1$ is prime. Let's look for less trivial solutions... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Two algebra questions I have two questions that I need help with:
1) How many single digit even natural number solutions are there for the equation $A+B+C+D = 24$ such that $A+B > C+D$
A)20 B)11 C)16 D)24
2) Three positive real numbers $x,y,z$ are such that $x+y+Z = 1$. which of the following inequalities best discribe... | For the first question, you have on the one hand that $(A+B)+(C+D)=24$ and on the other hand that $A+B>C+D$. What are the possibilities for $A+B$ and $C+D$? Clearly you must have $A+B>12$ and $C+D<12$. If two single-digit natural numbers add up to $11$ or less, what are the possibilities? (Note that the answer will dep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Maximum value of the modulus of a holomorphic function I'm looking for the maximum value of the modulus of a holomorphic function, and I am getting a bit stuck.
The function is $$(z-1)\left(z+\frac{1}{2}\right)$$ with domain $\,|z| \leq 1\,$
Now, I know by the maximum modulus principle the max value will occur on the ... | $$f(z) = (z-1)(z+1/2) = z^2 - z/2 - 1/2$$ Since you know that the maximum is hit on the boundary, $z = e^{i\theta}$, we get that $$F(\theta) = e^{2i \theta} - \dfrac{e^{i\theta}}2 - \dfrac12 = \left(\cos(2\theta) - \dfrac{\cos(\theta)}2 - \dfrac12 \right)+i \left(\sin(2 \theta) - \dfrac{\sin(\theta)}2\right)$$
Let $g(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\int^1_0\frac{x^2}{4x+5}dx$ How can I integrate this function ?
$$\int^1_0\frac{x^2}{4x+5}dx$$
| Divide the $x^2$ by $4x+5$, using ordinary division of polynomials. We get
$$\frac{x^2}{4x+5}=\frac{1}{4}x-\frac{5}{16}+\frac{25}{16}\frac{1}{4x+5}.$$
Now the integration should be straightforward.
Alternately, as a second choice, let $u=4x+5$. Then $du=4\,dx$, so $dx=\frac{1}{4}du$. Also, $x=\frac{1}{4}(u-5)$, so $x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find $\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$ $$\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$$
I have tried multiplying by $\frac{1}{\sqrt{x^2+4}}$ and it's reciprocal, but I cannot seem to find the solution. L'Hospital's doesn't seem to work either, as I keep getting rational square roots.
| $\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$=$\lim\limits_{x \to \infty} \frac{\sqrt{x^2(1 + \frac{4}{x^2})}}{x+4}$=$\lim\limits_{x \to \infty} \frac{x\sqrt{1 + \frac{4}{x^2}}}{x+4}$.
When $ x\longrightarrow\propto $ $\Rightarrow$ $\frac{4}{x^2}$$\longrightarrow$ 0 .
The above integral then takes the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$ $X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$
I changed the bases to 10, then performed manual addition/multiplication but it didn't yield me any result except for long terms. Please show me the way.
All I'm getting is $$\frac{\lg 18\lg54 +... | Note that $XY + 5(X - Y) = (X - 5)(Y + 5) + 25$, so it suffices to find $(X - 5)(Y + 5)$.
$(X - 5) = \log_{12}(18) - 5 = \log_{12}{18 \over 12^5} = \log_{12}{3^{-3}2^{-9}} = -3\log_{12}(24)$.
$(Y + 5) = \log_{24}(54) + 5 = \log_{24}(54*24^5) = \log_{24}(2^{16}3^{8}) = 8\log_{24}(12)$.
Multiplying together gives $-24\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
All integer solutions for $x^4-y^4=15$ I'm trying to find all the integer solutions for $x^4-y^4=15$.
I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, x^2+y^2=5$.
Only the last one is valid. $x^2+y^2=15$ is not solvable since the p... | First, assume $x$ and $y$ are strictly positive.
$x^4 - y^4 = (x-y)(x+y)(x^2+y^2)$. Of these three factors, only the first can equal 1. So they must be 1, 3, and 5. From 1 and 3 you already get $x=2, y=1$, and then you just have to check that this is a valid solution.
Now we allow the negative solutions, to get $x = \p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/166070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Recurrence relation $C_n = n + 1 + \dfrac{2}{n}\sum\limits_{k=0}^{n-1}C_k$. A Discrete Mathematics book from which I'm self-studying ("Discrete Mathematics and Its Applications", by Kenneth Rosen) asks me to do the following:
Given the following recurrence relation:
$$C_n = n + 1 + \frac{2}{n}\sum_{k=0}^{n-1}C_k$$
The ... | A simpler method :
$$n C_n = n(n + 1) + 2\sum_{k=0}^{n-1}C_k$$
$$(n-1) C_{n-1} = (n-1)n + 2\sum_{k=0}^{n-2}C_k$$
so that the difference is :
$$n C_n-(n-1) C_{n-1} = 2n+ 2C_{n-1}$$
that should give you the modified recurrence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/168214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Turn fractions into $\mathbb Z_7$ elements I had to perform a division between two polinomials $2x^2+3x+4$ and $3x+4$, my book suggests to do this operation without worrying about the modulo. So my result is $(3x+4)(\frac{2}{3}x+\frac{1}{9})+\frac{32}{9}$. Unfortunately my book fails to explain how should I perform the... | Remember that $\frac{1}{x}$ means "the element which, when multiplied by $x$, gives $1$."
Alternatively, $\frac{a}{b}$ is just shorthand for "the solution to the equation $bx=a$."
So, in $\mathbb{Z}_7$, the fraction $\frac{1}{3}$ is shorthand for "the solution to the equation $3x=1$", which, when interpreted in $\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Finding all primes $p$ such that $\frac{(11^{p-1}-1)}{p}$ is a perfect square How to Find all primes $p$ such that $\dfrac{(11^{p-1}-1)}{p}$ is a perfect square
| If $p=2$, $\frac{11^{p-1}-1}{p}$ is not a perfect square. In the following, we assume that $p\neq 2$.
Suppose that $\frac{11^{p-1}-1}{p}$ is a perfect square and write $p=2k+1$.
$$11^{2k}-1 = (11^k-1)(11^k+1)= p n^2 $$
Now, as $gcd(11^k-1, 11^k+1)=2$, we have one of the following cases
$$ 11^k-1 = p a^2\text{ and }11^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Find the remainder for $(x^5-x^4+x^3+2x^2-x+4)/(x^3+1)$
Find the remainder for $\dfrac{x^5-x^4+x^3+2x^2-x+4}{x^3+1}$
I know exactly how to synthetically divide in the format of: $(x\pm a)$. But not $(x^n\pm a)$ (with an exponent). So if anyone can tell me if anything changes or if the steps are the exact same just ... | It works like any standard division you've done. You start by taking the highest power of $x$ possible to substract to the numerator. In your case, your first term to be $x^2$. You keep on going until you are done, like any usual division. so for the first step you get
$$
\frac{x^5-x^4+x^3+2x^2-x+4}{x^3+1}=x^2+\frac{-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Evaluate :$\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx$ How to evaluate
$$
\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx
$$
I know that $\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}=\frac{\pi}{2}$ but after that I have no idea, ... | Let $$ I_0=\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx $$$$\Rightarrow I_0=\int\frac{\frac{\pi}{2}-2\cos^{-1}\sqrt{x}}{\frac{\pi}{2}}dx$$$$\Rightarrow I_0=\int (1-\frac{4}{\pi}\cos^{-1}\sqrt{x})dx$$ $$\Rightarrow I_0=x-\frac{4}{\pi}\int\cos^{-1}\sqrt{x})dx$$ Now Conside... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Find the integer solution of $ a^b = 2^{2 c + 1} + 2^c + 1 $ Find the possible number of integer solution for this equation, such that $ b>1$
$$ a^b = 2^{2 c + 1} + 2^c + 1 $$
From $1$ to $1000$, $ {a = 2, b = 2, c=0} $ and $ {a = 23, b = 2, c=4} $ computationally. Are there any other possible solutions? How to show an... | Assume $c>2$ for simplicity's sake, so that $a$ is odd among other things.
Firstly, we note that $a^b\equiv1$ mod $2^c$. Since the order of the multiplicative group mod $2^c$ is $2^{c-1}$, we note that either $b$ is even or that $a\equiv1$ mod $2^c$, but we can rule out the latter fairly easily - $1+2*2^c$ is already t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
| As I commented, you can exploit the symmetry.
$$\eqalign{
& f(1 - x) + {x^2}f(x) = 2x - {x^4} \cr
& f(x) + {\left( {1 - x} \right)^2}f(1 - x) = 2\left( {1 - x} \right) - {\left( {1 - x} \right)^4} \cr} $$
Now you get
$$\eqalign{
& f(1 - x) + {x^2}f(x) = 2x - {x^4} \cr
& f(1 - x) = \frac{{2\left( {1 - x} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
} |
Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$ Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$.
Can anyone help me with this? Thank You!
| Notice that for $\theta = 20, 40,$ and $80$ degrees you have $\tan^2(3\theta) = 3$. The tangent triple angle formula, which you can get from the tangent angle addition formula, says that
$$\tan(3\theta) = {3\tan(\theta) - \tan^3(\theta) \over 1 - 3 \tan^2(\theta)}$$
So the equation $\tan^2(3\theta) = 3$ can be expresse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 0
} |
Prove $\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)$ if $A+B+C=180$ degrees I most humbly beseech help for this question.
If $A+B+C=180$ degrees, then prove
$$
\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)
$$
I am not sure what trig identity I should use to begin this problem.
| $\sin^2A+\sin^2B-\sin^2C=\sin^2A+\sin(B+C)\sin(B-C)$
either using the identity $\sin^2B-\sin^2C=\sin(B+C)\sin(B-C)$
or $\sin^2B-\sin^2C=\frac{1}{2}(2\sin^2B-2\sin^2C)=\frac{1}{2}(1-\cos2B-(1-\cos2C))=\frac{1}{2}(\cos2C-\cos2B)=-\frac{1}{2}2\sin(B+C)\sin(C-B)=\sin(B+C)\sin(B-C)$
as $\sin(-x)=-\sin(x)$
Now, $\sin(B+C)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
} |
Prime numbers $(x,c,p)$ such that $x^3-p x^2-cx-5c=0$ How should I proceed to find all prime numbers $x,c,p$ such that
$$x^3-px^2-cx-5c=0$$
| If $x^3-px^2-cx-5c=0$, then $x$ divides $5c$. Since $x$ and $c$ are prime, we have the two possibilities $x=c$ and $x=5$.
Suppose $x=c$. Substitute. We get $c^3-(p+1)c^2-5c=0$, so $c^2-(p+1)c-5=0$, so $c$ divides $5$, so $c=5$. Therefore definitely $x=5$.
Substitute. We get $125-25p-10c=0$. Since $25$ divides $10c$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/179737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$? Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
| $\sqrt{a} = x, b=y^2, c=z^2 => x^2+y^2+z^2=1$
We have to prove $$\frac{x}{y^{2}+z^{2}}+\frac{y}{x^{2}+z^{2}}+\frac{z}{x^{2}+y^{2}}\geq \frac{3\sqrt{3}}{2}$$:
$$\frac{2\sqrt{3}}{3}x\left ( y^{2}+z^{2} \right )\leq \left ( x^{2}+\frac{1}{3} \right )\left ( y^{2}+z^{2} \right )\leq \frac{\left ( x^{2}+y^{2}+z^{2}+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Prove that $ \frac{n}{\phi(n)} = \sum\limits_{d \mid n} \frac{\mu^2(d)}{\phi(d)} $ I am trying to show that:
\begin{equation}
\frac{n}{\phi(n)} = \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)}
\end{equation}
where $\phi(n)$ is Euler's totient function and $\mu$ is the Möbius function.
The identity clearly holds for $n=1$, so... | $$\sum_{d|n}\frac{\mu^2(d)}{\phi(d)}=1+\frac{1}{p_1-1}+\frac{1}{p_2-1}+\cdots+\frac{1}{(p_1-1)(p_2-1)+\cdots}$$
$$=(1+\frac{1}{p_1-1})(1+\frac{1}{p_2-1})\cdots=\prod_{p_i|n}(1+\frac{1}{p_i-1})=\prod_{p_i|n}(\frac{1}{1-1/p_i})
$$
$$
=\frac{n}{\phi(n)}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/182720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
How to combine ratios? If $a:b$ is $2:5$, and $c:d$ is $5:2$, and $d:b$ is $3:2$, what is the ratio $a:c$? How would I go about solving this math problem?
if the ratio of $a:b$ is $2:5$ the ratio of $c:d$ is $5:2$ and the ratio of $d:b$ is $3:2$, what is the ratio of $a:c$?
I got $\frac{a}{c} = \frac{2}{5}$ but that is... | Divide a:b with c:d
i.e. $$\frac{(\frac{a}{b})}{(\frac{c}{d})} = \frac{(\frac{2}{5})}{\frac{5}{2}}$$
$\implies$ $\frac{a}{c}*\frac{d}{b} = \frac{4}{25}$
$\implies$ $\frac{a}{c}*\frac{3}{2} = \frac{4}{25}$
$\implies$ $\frac{a}{c}= \frac{8}{75}$
Now talking about the approach, So you must try to figure the ratio whose v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
is there a method to find all the solutions to, $y^2-6y+\sqrt{y}+4=0$ is there a method to find all the solutions to the following set of irrational equations,
$\sqrt{x}+y=3$
$x+\sqrt{y}=5$
NOTE: $(4-1)=(2-1)(2+1)=3$ and $(4+1)=(2^2+1^2)=(3^2-2^2)=(3-2)(3+2)=5$
| If you make the change of variables
$$\begin{equation*}
u=\sqrt{x}\geq 0,\qquad v=\sqrt{y}\geq 0,
\end{equation*}$$
then you need to solve
$$\begin{equation*}
\left\{
\begin{array}{c}
u+v^{2}=3 \\
u^{2}+v=5
\end{array}
\right.
\end{equation*}$$
or
$$\begin{eqnarray*}
&&\left\{
\begin{array}{c}
u=3-v^{2} \\
v^{4}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Writing 1/3 as a sum of other numbers Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it should be an infinite series, but I can be wrong.
In case if it can't be written usi... | There is no way to write it as a finite sum. For if you bring such a sum to a common denominator, that denominator will be a power of $2$. Minus signs won't help.
It can be expressed as the infinite "sum"
$$\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}+\cdots.$$
For note that if $|r|\lt 1$ the geometric series ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
how can one solve for $x$, $x =\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}$
Possible Duplicate:
Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$
how can one solve for $x$, $x =\sqrt[]{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}}}}$
we know, if $x=\sqrt[]{2+\sqrt{2}}$, then, $x^2=2+\sqrt{2}$
now, if $x=\... | Well, $x = (2 + x)^.5 $ from the expression of x. Implying $x^2 = 2 + x$. And now you can solve the quadratic for x, giving x = -1 and x = 2. X can't be negative, so x = 2
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/186652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
A cube is divided into two cuboids A cube is divided into two cuboids. The surfaces of those cuboids are in the ratio $7: 5$. Calculate the ratio of the volumes.
How can I calculate this?
| Without loss of generality we can let the sides of the original cube be $1$.
The larger of the two cuboids has four of its sides equal to say $x$. (The other $8$ sides are $1$.) Then the smaller cuboid has four of its sides equal to $1-x$.
The surface area of the larger cuboid is $2+4x$. (Two $1\times 1$ faces, and fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Permutation across matrices. Matrices may be used to permute the order of elements in a set. For example:
$$
\begin{bmatrix}
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z \\
... | There's a pretty good chance I don't know what you're asking, but maybe the matrix you're looking for is $$M=\pmatrix{0&1&0&0\cr0&0&1&0\cr0&0&0&1\cr1&0&0&0\cr}$$ You get $$MA=\pmatrix{0&1&0&0\cr0&0&1&0\cr0&0&0&1\cr1&0&0&0\cr}\pmatrix{0&0&x&0&0\cr x&x&x&x&x\cr x&0&0&0&x\cr0&0&0&0&0\cr}=\pmatrix{x&x&x&x&x\cr x&0&0&0&x\cr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
All pairs (x,y) that satisfy the equation $xy+(x^3+y^3)/3=2007$ How we can find the all pairs $(x,y)$ from the integers numbers ,that satisfy the equation :
$$xy+\frac{x^3+y^3}{3} =2007$$
| Note that the given equation is $$x^3+y^3+3xy=6021$$ or $$x^3+y^3-1+3xy=6020$$
Factoring it we get , $$(x+y-1)(x^2+y^2+1+x+y-xy)=2^2.5.7.43$$
Obviously check $\equiv 3$ and see $$x+y-1\equiv 2 \mod 3$$
Also, $$x+y-1 < x^2+y^2+1+x+y-xy$$
This means, $$x+y-1 \rightarrow 20,5,2,35$$
so now it's easy to see that $$x+y-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Small math help with polynomials If one solution of the equation $3x^2 = 8x + 2k + 1$ is $7$ times the other. Find the solutions and the value of $K$.
Note: This isn't a homework question. I'm skipping ahead in my textbook.
Thank you for the help.
Also please help me with this question no.2 also.
Find the other zeroes ... | $\bigstar$ First question:
First we form the standard notation:$3x^2-8x-(2k+1)=0$
We know that the sum of the roots of a quadratic equation of the form $ax^2+bx+c$ is $x_1+x_2=-\frac{b}{a}$
So: $x_1+x_2 =\frac{8}{3}$
On the other hand: $x_1=7\times x_2$
Solving this two equations two unknowns yields:$x_2 = \frac13$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Inequality. $ab^2+bc^2+ca^2 \geq a+b+c.$ Using rearrangement inequalities prove the following inequality:
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Prove that
$$ab^2+bc^2+ca^2 \geq a+b+c.$$
Thanks :)
| I think I have the solution using arrangements inequalities.(source: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=497213)
We make the substitution $\displaystyle a=\frac{y}{x}, b=\frac{z}{y}, c=\frac{x}{z}$. We have now:
$$\frac{z^2}{xy}+\frac{x^2}{yz}+\frac{y^2}{xz} \geq \frac{y}{x}+\frac{z}{y}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Simple partial differentiation $x = r\cos\theta$ and $y = r\sin\theta$ If
\begin{align}
x &= r\cos\theta,\\
y &= r\sin\theta,
\end{align}
find
$$\dfrac{\partial^2\theta}{\partial{x}\partial{y}}.$$
How can I find this partial derivative?
I need to prove that
$$
\frac{\partial^2\theta}{\partial{x}\partial{y}} = -\frac... | Using $\theta = \arcsin\left(\frac{y}{r}\right)$ and $\arcsin'(t) = \frac{1}{\sqrt{1-t^2}}$ we get
$$ \frac{\partial \theta}{\partial y} = \frac{1}{r} \frac{1}{\sqrt{1- \left(\frac{y}{r}\right)^2}} = \frac{1}{x}$$
using $r^2 = x^2 + y^2$. Hence
$$\frac{ \partial \theta}{\partial x\partial y} = -\frac{1}{x^2} = -\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
The value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$ There was a multiple choices saying:
Find the value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$ knowing that $x^2+... | As $x^2+x<0, (x-0)(x-(-1))<0$
Now the product two terms is negative, so
either ($x-0>0$ and $x-(-1)<0$) or ($x-0<0$ and $x-(-1)>0$).
If $x>0$ and $x<-1\implies -1>x>0\implies -1>0$, which is clearly impossible.
If $x<0$ and $x>-1$, $-1<x<0$
$\implies -1<x^{2m+1}<0\implies \bigl\lfloor x^{2m+1}\bigr\rfloor=-1$
$\imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Accuracy from approximating $\zeta(2)$ with a partial sum This is for an introductory numerical analysis class. The answer shouldn't be too complicated, but if you have one, feel free to post it.
Figure out what $n$ should be such that
$$\sum_{k=n+1}^\infty {1\over k^2}<10^{-8}.$$
My Simple Algebraic Attempt
We know t... | Since $\displaystyle \frac{1}{k^2} < \frac{1}{(k-1)k } = \frac{1}{k-1} - \frac{1}{k}$ we can bound your term by a telescoping sum: $$\sum_{k=n+1}^{\infty} \frac{1}{k^2} < \left(\frac{1}{n} - \frac{1}{n+1} \right)+\left(\frac{1}{n+1} - \frac{1}{n+2} \right)+\left(\frac{1}{n+2} - \frac{1}{n+3} \right) + \cdots = \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Expressing in the form $A \sin(x + c)$
Express in the form $A\sin(x+c)$
a) $\sin x+\sqrt3\cos x$; b) $\sin x-\cos x$
sol: a) $A=\sqrt{1+3}=2$, $\tan c=\frac{\sqrt 3}1$, $c=\frac\pi3$. So $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$
b) $\sqrt 2\sin(x-\frac\pi4)$
Can someone please explain the method used in the provi... | The anonymous commenter has already answered your question, but in case you have any remaining doubts, I will provide a detailed answer.
For starters, do note that
$A \, \sin (x + c) = \left(A \, \cos(c) \right) \, \sin(x) + \left( A \, \sin(c)\right) \, \cos(x)$
Since you have $\sin (x) + \sqrt{3} \, \cos (x)$, it fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Prove $ 1 + 2 + 4 + 8 + \dots = -1$
Possible Duplicate:
Infinity = -1 paradox
I was told by a friend that $1 + 2 + 4 + 8 + \dots$ equaled negative one. When I asked for an explanation, he said:
Do I have to?
Okay so, Let $x = 1+2+4+8+\dots$
$2x-x=x$
$2(1+2+4+8+\dots) - (1+2+4+8+\dots) = (1+2+4+8+\dots)$
Therefore, ... | It is similar to "proving" that 1 = 2 by saying that 1+infinity = 2+infinity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
} |
Least Common Multiple of 3 modulo equations i%7=1 -> i=1+7n
i%9=2 -> i=2+9n
i%11=3 -> i=3+11n
Besides writing out everything:
1,8,15,22...
2,11,20,29...
3,14,25,36...
to find the LCM. Is there a better way to do this?
| In general, you should use the Chinese Remainder Theorem.
However, for these particular numbers, there is a trick! Note that the congruences are equivalent to $2x+5\equiv 0\pmod{7}$, $2x+5\equiv 0\pmod{9}$, and $2x+5\equiv 0\pmod{11}$. So we want $2x+5$ to be divisible by $7$, $9$, and $11$, and therefore by $(7)(9)(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
List the primes for which the following system of linear equations DOES NOT have a solution in $\mathbb{Z}_p$ Let $p$ be a prime and consider the field $\Bbb Z_p$. List the primes for which the
following system of linear equations does not have a solution in $\Bbb Z_p$:
$$
\begin{align}
5x + 3y &= 4 \tag{1}\\
3x + 6y &... | Form the extended coefficients matrix and apply Gauss Reduction as much as possible.
To begin with, if $\,p=5\,$ then the first equation is $\,3y=4\Longrightarrow y=4\cdot 3^{-1}=4\cdot 2=3\,$ , and substituing in eq. 2 we get $\,3x+3=1\Longrightarrow 3x=-2=3\Longrightarrow x=1\,$ , so there's a unique solution: $\,(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$(1+i)$ to the power $n$
Possible Duplicate:
Complex number: calculate $(1 + i)^n$.
I came across a difficult problem which I would like to ask you about:
Compute
$ (1+i)^n $ for $ n \in \mathbb{Z}$
My ideas so far were to write out what this expression gives for $n=1,2,\ldots,8$, but I see no pattern such that I ca... | Use: $$\cos(n\theta)=\cos^n(\theta)-\frac{n(n-1)}{2!}\cos^{n-2}(\theta)\sin^2(\theta)+\frac{n(n-1)(n-2)(n-3)}{4!}\cos^{n-4}(\theta)\sin^4(\theta)-...$$ and $$\sin(n\theta)=n\cos^{n-1}(\theta)\sin(\theta)-\frac{n(n-1)(n-2)}{3!}\cos^{n-3}(\theta)\sin^3(\theta)-...$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/199004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Erased number? A set of consecutive positive integers starting with 1 is written on the board. A student came along and erased one number. Average of remaining numbers is 61 15/20 . What was the number erased
| Let $n$ be the last number written. Lets say that $m$ is the erased number.
Then the sum of the numbers on the board is $\frac{n(n+1)}{2}-m$. Their average then is
$$\frac{\frac{n(n+1)}{2}-m}{n-1}=61 \frac{15}{20}$$
Multiplying by 2 you get
$$\frac{n(n+1)-2m}{n-1}=122\frac{3}{2}$$
$$\frac{n^2+n-2}{n-1}+\frac{2}{n-1}-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Multiplicative inverse of a quadratic algebraic number $\,a+b\sqrt 2$ Find the multiplicative inverse of $1+ 3\sqrt{2}$ in the ring $\mathbb{Q}(\sqrt{2})$ and use it to solve the equation $(1+3\sqrt{2})x=1-5\sqrt{2}$.
I think that the inverse is the conjugate, so it would be $1-3\sqrt{2}$, but then I don't know where t... | Instead of finding inverse, you can directly find(less computation) $x$
Let $x=a+b\sqrt 2$
Then, $(1+3\sqrt 2)(a+b\sqrt 2)=1-5\sqrt 2\implies (a+6b)+(3a+b)\sqrt 2=1-5\sqrt 2$
$\implies a+6b=1$ and $3a+b=-5$.
Solving these equations, we get
$a=-\frac{31}{17}$ and $b=\frac{8}{17}\implies x=-\frac{31}{17}+\frac{8}{17}\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $ If we know that for every $a,b,c>0$ ,how we can find the least possible value for :
$$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $$
| Put $(a,b,c)=(3,4,4)$ to get 4. I will show this is optimal.
Assume without loss of generality that $a \leq b \leq c$. Two cases:
If $c \geq a+b$ then $\lfloor\frac{c+a}{b}\rfloor \geq 1$ and $\lfloor\frac{c+b}{a} \rfloor \geq \lfloor\frac{a+b+b}{a} \rfloor \geq 3$ so the sum is at least 4.
If $c \leq a+b$ then $\lfloo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
How do we solve the equation? How do we solve the following equation in the set of real numbers?
$$(x+1)\cdot \sqrt{x+2} + (x+6)\cdot \sqrt{x+7}=(x+3)\cdot (x+4).$$
I wrote the given equation has the form
\begin{equation*}
(x+1)(\sqrt{x + 2} - 2) + (x + 6)(\sqrt{x+7} - 3) = (x-2)(x+4)
\end{equation*}
This equation i... | Not much simpler, but you can also try something like $a:=\sqrt{x+2}$ then $\sqrt{x+7}=\sqrt{a^2+5}$, then it will have only one sqrt in the equation, put that on one side and the rest on the other side..
$$(a^2-1)a+(a^2+4)\cdot\sqrt{a^2+5} = (a^2+1)(a^2+3)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/202670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Chebyshev Diff EQ Find a power series solution about $x_0=0$ for the Chebyshev differential equation
$$(1-x^2)y''-xy'+n^2 y=0,$$
as a function of of the integer $n$. Show that the solutions form a terminating expansion for each value of $n$. What is the orthogonality relationship for these polynomials?
| The power series around zero is
$$y(x) = \sum_{k=0}^\infty a_k \,x^k.$$
Therefore
$$
y' = \sum_{k=0}^\infty k \,a_{k} \,x^{k-1} =\sum_{k=1}^\infty k \,a_{k} \,x^{k-1}=\sum_{k=0}^\infty \,(k+1) \,a_{k+1} \,x^{k},
$$
and
$$
y'' = \sum_{k=0}^\infty \, k\,(k+1) \, a_{k+1} \,x^{k-1}= \sum_{k=1}^\infty \, k\,(k+1) \, a_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Ideas on the ways to integrate $\int \tan^2( x)\sec^3( x) dx$ I would proceed by thus , let $y = [\sec (x)]^2 $
then
$$dy = 2 \cdot \sec(x) \cdot \sec(x) \cdot \tan(x) \cdot dx = 2 \cdot ( \sec (x))^2 \cdot \tan(x) \cdot dx $$
so,
$$
2 \tan^2(x) \sec^2 (x) dx = \sec(x) \cdot \tan(x) \cdot dy = y(y-1)^\frac{1}{2} \cdo... | We can do this by intgration by parts
$ I=\int tan^2 x \cdot sec^3x \space dx$
$=\int (sec^2 x-1)\cdot sec^3 x \space dx$
$=\int sec^5 x \space dx-\int sec^3 x \space dx$
$=sec^3 x \cdot tan x-\int 3sec^2 x \cdot sec x tan x \cdot tan x \space dx-\int sec^3 x \space dx $
$= sec^3x \cdot tanx-3I-I_1$
$ or, 4I= sec^3x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.
Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.
I thought the answer is $1+1*2+2*2^2+3*2^3+4*2^4+5*2^5+6*2^6+7*2^7+8*2^8+9*2^9+2^{10}=9219$, but the answer should be 8204. What mistake have I made?
| We have $\lfloor \log_2 n \rfloor = k$ iff $2^k \le n < 2^{k+1}$, so for $0 \le k \le 9$, $k$ appears in the above sum exactly $2^{k+1} - 2^k = 2^k$ times. As $10$ appears exactly once (for $n =1024$), we have
\[
\sum_{n=1}^{1024} \lfloor \log_2 n \rfloor = 10 + \sum_{k=0}^9 k2^k
= 8204.
\]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/203905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving recurrence relation in 2 variables We already know how to solve a homogeneous recurrence relation in one variable using characteristic equation. Does a similar technique exists for solving a homogeneous recurrence relation in 2 variables. More formally, How can we solve a homogeneous recurrence relation in 2 va... | Use generating functions like the one variable case, but with a bit of extra care. Define:
$$
G(x, y) = \sum_{r, s \ge 0} F(r, s) x^r y^s
$$
Write your recurrence so there aren't subtractions in indices:
$$
F(r + 1, s + 1) = F(r + 1, s) + F(r, s + 1)
$$
Multiply by $x^r y^s$, sum over $r \ge 0$ and $s \ge 0$. Recognize... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 4,
"answer_id": 2
} |
What is the pattern or relation in this table? Here is the table:
$$\begin{array}{c}
0\\
1\\
1& 1\\
3& 2& 3\\
5& 3& 3& 5\\
11 &8 &10 &8 &11\\
21 &13 &14 &14 &13 &21\\
43 &30 &37 &36 &37 &30 &43\\
85 &55 &61 &55 &55 &61 &55 &85\\
171 &116 &140 &140 &146 &140 &140 &116 &171
\end{array}$$
In this trian... | Updated:
let $A(n,k)$ be the number in the table with row $n$ and column $k$, where $0\le n$ and $0\le k\le n$.
$$A(n,k)=A(n,n-k)$$
$$A(n,0)=A(n,n)=0$$
The table would look like this: (top left is row $0$ and column $0$)
$$\begin{array}{c}
0\\
0& 0\\
0& 1& 0\\
0& 1& 1& 0\\
0& 3& 2& 3& 0\\
0& 5& 3& 3& 5& 0\\
0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Find the equation of the plane that contains Find the equation of the plane that contains,
the lines: $$
\frac{x-2}{2} = \frac{y+4}{3} = \frac{2 - z}{5}
$$
and
$$\begin{align}
x &= 3 + 4t \\
y &= -4 +6t \\
z &= 5 -10t.
\end{align}
$$
I'm not exactly sure on how to tackle this problem.
| The line $\frac{x-2}{2} = \frac{y+4}{3} = \frac{2-z}{5}\,$ can be written as
$$x=2+2t\,,\quad y=-4+3t \,,\quad z=2-5t \,, $$
and the line
$$x = 3 + 4t \,,y = -4 + 6t \,,z = 5 - 10t \,.$$
The two lines have the same direction, since $ v_1=(2,3,-5) $ and $ v_2 = (4, 6,-10)\,, $ where $v_1$ and $v_2$ are the direction ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Transformation matrix to go from one vector to another I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?
| Try using the dyadic product, the definition is
$$
\mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Generating Functions help Using generating functions, find the number of ways to make change for a $\$100$
bill using only dollar coins and $\$1,\$2$, and $\$5$ bills.
Thanks for the help.
| Your generating function will have the form $$f(x)=\sum_{n\ge 0}a_nx^n\;,$$ where $a_n$ is the number of ways to make a total of $n$ dollars using the prescribed coin and bills. Each $\$1$ coin must therefore add $1$ to the exponent, as must each $\$1$ bill; each $\$2$ bill must add $2$ to the exponent, and each $\$5$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/210404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solve for $x$: $\big(x^3+\frac{1}{x^3}+1\big)^4=3\big(x^4+\frac{1}{x^4}+1\big)^3$ Solve for $x$
$$\big(x^3+\frac{1}{x^3}+1\big)^4=3\big(x^4+\frac{1}{x^4}+1\big)^3$$
let $x+\frac{1}{x}=t$ the equation equivalent to $(t^3-3t+1)^4=3(t^4-4t^2+3)^3$
but it's very complicated. Thanks.
| Let $u=t^4-4t^2+3$ and $v=t^3-3t+1$. To finish off the problem (over real numbers), it suffices to prove that the equation $3u^3=v^4$ has no solutions with $|t|\ge 2$ other than $t=2$. Differentiate the ratio $f(t)=3u^3 v^{-4}$:
$$ f'(t)=\frac{d}{dt}(3u^3v^{-4})= 3u^2v^{-5}(3u'v-4uv') \tag1$$
The sign of $v$ is the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/216935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Nonlinear equations and unique solution How to show that the following system of equations has a unique solution $(x,y)$? $x,y$ are scalars.
$x+\frac{3}{4}y+\frac{1}{20}\sin x=0$
$-\frac{37}{40}x+y+\frac{1}{10}\sin y=0$
I tried contraction mapping, but it didn't work.
| I solved the top equation for $y$, substituted it into the second equation, and got
$$\frac{1}{10}\sin(\frac{1}{15}\sin x + \frac{4}{3}x) + \frac{271}{120}x + \frac{1}{15}\sin x = 0$$. One solution is $x = 0$, as you found. The derivative of the left-hand side is always positive:
$$\frac{1}{10}\cos(\frac{1}{15}\sin ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proving a sum using induction I am having a problem with this question.
I need to prove by induction that: $$\sum_{k=1}^n \sin(kx)=\frac{\sin(\frac{n+1}{2}x)\sin(\frac{n}{2}x)}{\sin(\frac{x}{2})}$$
The relation is obvious for n=1
Now I suppose that the relation is true for a natural number n and I want to show that $$\... | You can actually use telescopy, which is just induction in disguise.
We have that
$$\cos b - \cos a = 2\sin \frac{{a + b}}{2}\sin \frac{{a - b}}{2}$$
Now let $$b=\left(k+\frac 1 2 \right)x$$
$$b=\left(k-\frac 1 2 \right)x$$
Then
$$\cos \left( {k + \frac{1}{2}} \right)x - \cos \left( {k - \frac{1}{2}} \right)x = 2\sin k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
As shown in the figure: Prove that $a^2+b^2=c^2$ Geometry: Buildings in the triangle
Other triangles with the same property:
$1.$ 12 18 6 12 30 102
$2.$ 15 30 15 15 15 90
$3.$ 24 30 54 24 6 42
$4.$ 30 10 40 30 20 50 (proposed problem in sense of clockwise)
$5.$ 36 12 6 12 18 96
$6.$ 36... | I add the letters to your points
Using Theorem of Sine, we get
$$\frac{a}{\sin 30^\circ}=\frac{BD}{\sin(10^\circ+40^\circ)}=\frac{BD}{\sin 50^\circ}$$
$$\frac{BD}{\sin 40^\circ}=\frac{BA}{\sin(30^\circ+20^\circ+50^\circ)}=\frac{BA}{\sin 100^\circ}$$
so we get
$$a=\left(\frac{BA\cdot\sin 30^\circ}{\sin 100^\circ\sin 50... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 0
} |
How do I solve this equation How do I solve this equation, I keep getting the wrong answer.
$$ \frac{-2x^2+50}{\sqrt{50-x^2}} = 1 $$
Thank you
| We have:
$$\sqrt{50 - x^2} = 50 - 2x^2$$
Eliminating the square root and opening the brackets:
$$50 - x^2 = 4x^4 - 200x^2 + 2500$$
Make it more simple:
$$-4x^4 + 199x^2 - 2450 = 0$$
Substitute $y = x^2$, simplifying and find full square:
$$\left(y - \dfrac{199}{8}\right)^2 = \dfrac{401}{64}$$
So, there is two solutions... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove the following relation: I must prove the relation $$\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k}=2\sum_{k=0}^n\binom{n+k}{k}\frac1{2^k}.$$
I got this far before I got stuck:
$\begin{eqnarray*}
\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k} & = & \sum_{k=0}^{n+1}\left\{\binom{n+k}{k}+\binom{n+k}{k-1}\right\}\frac1{2^k... | You correctly used Pascal's identity, but then you goofed going to the next line. (Should have an $n$ in that last exponent of $2$, not a $k$.) I recommend going a different way, though.
$\begin{eqnarray*}
\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k} & = & \sum_{k=0}^{n+1}\left\{\binom{n+k}{k}+\binom{n+k}{k-1}\right\}\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Probability that I will get a $4$ consecutive numbers I have five dice to roll. I roll them. What is the probability that I will get a straight with exactly four consecutive numbers and not $5$?
There are three options: $1,2,3,4$ or $2,3,4,5$ or $3,4,5,6$.
I have $1,2,3,4,*$ where $*$ can be either $1/2/3/4/6$. It cann... | As you have noticed, there are $14$ possible multisets of $5$ numbers that are "good" in your sense, of containing four consecutive numbers but no five consecutive numbers. These $14$ multisets can be divided into two kinds:
*
*$12$ of them with some number repeated, namely: $\{1, 2, 3, 4, 1\}, \{1, 2, 3, 4, 2\}, \{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Trig Identities : $\frac{\sin (4x)}{1-\cos(4x)} \frac{1-\cos(2x)}{\cos(2x)} = \tan(x)$ I want to prove that $$\frac{\sin (4x)}{1-\cos(4x)} \frac{1-\cos(2x)}{\cos(2x)} = \tan(x)$$
\begin{align}
\text{Left hand side} : & = \sin(2x+2x)/(1-\cos(2x+2x)) \times ((1-\cos^2x+\sin^2x)/(\cos^2x-\sin^2x))\\
& = ((2\sin^2x)(\cos^2... | Recall the following identities:
$$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$$
$$1-\cos(2 \phi) = 2 \sin^2(\phi)$$
Make use of the above identities and you will get your solution.
Move your cursor over the gray area for complete solution.
\begin{align}\dfrac{\sin(4x)}{1-\cos(4x)} \dfrac{1 - \cos(2x)}{\cos(2x)} & = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proving a limit with the epsilon-delta definition I was looking to prove using the $\epsilon,\delta$ limit definition that $\lim_{x\to a}(\sqrt[3]{x})=\sqrt[3]{a}$, $(a>0)$. I'm not sure what sort of algebraic manipulation I should use on the expression $|\sqrt[3]{x}-\sqrt[3]{a}|$ (so I'll be able to continue with prov... | You can use the identity
$$
x - a = \left( \sqrt[3]{x} - \sqrt[3]{a} \right) \left( \sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot \sqrt[3]{a} + \sqrt[3]{a^{2}} \right),
$$
which is derived from the following identity:
$$
a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}).
$$
Then
$$
\forall x \in \mathbb{R} \setminus \{ a \}: \quad \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Geometrical Inequalities I couldnt solve the following: we need to minimize $$\sqrt{\frac{(a+b-c)(b+c-a)(a+c-b)}{(a+b+c)}},$$ where a,b,c are sides of a triangle.
| This expression can be arbitrarily close to $0$. Let $a=2\varepsilon$, $b = c = 1/2 - \varepsilon$. There is a triangle with sides $a$, $b$, and $c$. As $\varepsilon$ goes to $0$, the expression approaches $0$. Specifically, we have
$$\sqrt{\frac{(a+b-c)(b+c-a)(a+c-b)}{(a+b+c)}}
= \sqrt{\frac{2\varepsilon \cdot (1 - 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$ If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$
I have tried to solve the equation first and find the values of the roots but it become... | Define $w_1=(1+\sqrt{3}i)/2$ and $w_2=(1-\sqrt{3}i)/2$ where $i=\sqrt{-1}$. Now observe that
\begin{align}
(a^3+1)=(a+1)(a-w_1)(a-w_2)
\end{align}
Similarily for other terms also. This helps you to write
\begin{align}
(a^3+1)(b^3+1)(c^3+1)(d^3+1)=P_1P_2P_3
\end{align}
where
\begin{align}
P_1&=(a+1)(b+1)(c+1)(d+1) \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/241751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
$2n+1$ and $n^2+1$ are always coprime or their gcd is $5$ Using a spreadsheet, it can be inferred that when $n≡2[5]$, then $\gcd(n^2+1,2n+1)=5$, else $\gcd(n^2+1,2n+1)=1$.
Indeed, when $n≡2[5]$, $n^2+1$ and $2n+1$ can easily be shown to be multiples of $5$, so their gcd is at least $5$. But then, I can't see how to com... | Let $d=\gcd(2n+1,n^2+1)$. Then since $d\mid 2n+1$ and $d\mid n^2+1$, we must also have $$d\mid(n^2+1)-(2n+1)=n(n-2)\;.$$ Clearly $\gcd(n,2n+1)=1$, and $d\mid 2n+1$, so $\gcd(n,d)=1$, and therefore $d\mid n-2$. By definition $d\mid 2n+1$, so $d\mid(2n+1)-2(n-2)=5$, and therefore $d=1$ or $d=5$.
The problem doesn’t requi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/242610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Limit of Functions Using $\lim_{n\to\infty} (1+\frac{x}{n}) ^n = e^x $ Can someone help me solve the following three exercices?
1) $\displaystyle \lim_{x\to \infty} \left( 1+ \frac{1}{9x^2 + x + \frac{1}{x} } \right)^{x^2 + \Large\frac{8}{x} } $
2) $\displaystyle \lim_ {x \to \infty} \left( 1+ \frac{1}{9x^4 + 8x^2 +8 ... | I'll show you one, but they are all very similar:
$$\begin{align*}\lim_{x\to \infty} \left( 1+ \frac{1}{9x^2 + x + \frac{1}{x} } \right)^{x^2 + \large \frac{8}{x} }&=\lim_{x\to \infty} \left(\left( 1+ \frac{1}{9x^2 + x + \frac{1}{x} } \right)^{9x^2+x+\Large \frac1x} \right)^{\Large \frac{x^2 + \Large\frac{8}{x}}{9x^2+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/245068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Limit $\lim_{n \rightarrow \infty}\frac{1 \cdot 3 \cdot 5 \dots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n)} $ $\displaystyle \lim_{n \rightarrow \infty}\frac{1 \cdot 3 \cdot 5 \dots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n)} $
| Note that
$$\frac{{1.3{\cdots}(2n - 1)}}{{2.4{\cdots}(2n)}} = \frac{{\frac{{(2n - 1)!}}{{2.4{\cdots}(2n - 2)}}}}{{{2^n}n!}} = \frac{{\frac{{(2n - 1)!}}{{{2^{n - 1}}(n - 1)!}}}}{{{2^n}n!}} = \frac{{(2n - 1)!}}{{{2^{n - 1}}(n - 1)!}} \cdot \frac{1}{{{2^n}n!}} \to 0$$
using Stirling's approximation because $n$ is "large" ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How to show that $f\left( \frac{ x + y }{ 2 }\right ) \leq \frac{ f( x ) + f( y ) }{ 2 }$ when $f''(x) \geq 0$. I need to show that if $f: (a,b) \to \mathbb{ R }\text{ with}\;\; f''( x ) \geq 0$ for all $x \in (a,b)$, then $f\left( \frac{ x + y }{ 2 } \right) \leq \frac{ f( x ) + f( y ) }{ 2 }$.
I know that since $f''(... | Since $f''(x) \geq 0$, $f'(x)$ is monotone increasing (as you already observed). Let $x, y \in (a, b)$ and WLOG $x < y$. Therefore, $x < \frac{x + y}{2} < y$. Now, applying MVT in $\left(x, \frac{x+y}{2}\right)$ and $\left(\frac{x + y}{2}, y\right)$, we get
$$
f'(c_1) = \frac{f\left(\frac{x + y}{2}\right) - f(x)}{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/248055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum?
Suppose that $x$, $y$, $x−y$, and $x+y$ are all positive prime numbers. What is the sum of the four numbers?
Well, I just guessed some values and I got the answer.
$x=5$, $y=2$, $x-y=3$, $x+y=7$. All the numbers are prime and the answer is $17$.
Supp... | If $x,y > 1$ are both odd, then $x+y > 2$ is even, a contradiction.
Then $y = 2$ and by inspection $x=5$ as $x = 3$ results in $x-y = 1$.
Or, we have that $x-2,x,x+2$ are all prime meaning we have three consecutive integers modulo $3$ which is only satisfied by the triple $(3,5,7) \implies y = 5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/250584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "100",
"answer_count": 5,
"answer_id": 4
} |
How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$? How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$ in the $C^2(0,\infty)$ space solutions?
| Expanding on Jack D'Aurizio's answer,
write the equation as
$f(x^2) = f''(x)$.
If $f(x) = \sum_{n=0}^{\infty} a_n x^n$,
$f(x^2) = \sum_{n=0}^{\infty} a_n x^{2n}$
and $f''(x) = \sum_{n=2}^{\infty} n (n-1)a_n x^{n-2}
= \sum_{n=0}^{\infty} (n+2) (n+1)a_{n+2} x^n
$
so $a_{2n} = (n+2) (n+1)a_{n+2}$
and $a_{2n+1} = 0$.
Sett... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
$\left(\sum_{j=0}^\infty\frac{z^j}{j!}\right)\left(\sum_{k=0}^\infty\frac{w^k}{k!}\right)=\sum_{n=0}^\infty\sum_{j=0}^n\frac{z^jw^{n-j}}{j!(n-j)!}$ I've been going through some series notes from my lecture and got stuck at this equality: $$\left(\sum_{j=0}^\infty\frac{z^j}{j!}\right)\left(\sum_{k=0}^\infty\frac{w^k}{k!... | $$
(1+2+3+4+\cdots)\cdot\left(\begin{array} {} & \text{one} \\[6pt] + & \text{two} \\[6pt] + & \text{three} \\[6pt] + & \text{four} \\[6pt] + & \cdots \end{array}\right)
$$
$$
= \sum \left[ \begin{array}{cccc} 1\cdot\text{one}, & 2\cdot\text{one}, & 3\cdot\text{one}, & 4\cdot\text{one}, & \cdots\\[6pt]
1\cdot\text{two... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
Find the radius of convergence of a power series expansion of the rational function $f(z)=\frac{(z^2)-1}{(z^3)-1 }$ I am having trouble figuring out the answer
| $$
\frac{z^2-1}{z^3-1} = \frac{(z-1)(z+1)}{(z-1)(z^2+z+1)} = \frac{z+1}{z^2+z+1}
$$
The denominator is $0$ when
$$z=\dfrac{-1\pm\sqrt{1^2-4\cdot1\cdot1}}{2} = \frac{-1\pm i\sqrt{3}}{2}. $$
If you want this in powers of $z$, i.e. powers of $(z-0)$, so the center is $0$, then the radius of convergence is the distance fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/257337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
show that if $a,b,c \in \mathbb{R}^+$ show that if $a,b,c \in \mathbb{R}^+$ different from zero, then:
$$(a^2+b^2+c^2)\cdot\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq(a+b+c)\cdot\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
I had no success in my attempts
| The direction of the inequality is flipped. Here is a proof of the correct direction.
Upon expansion we see it suffices to prove:
$$\frac{a^2}{b^2} + \frac{a^2}{c^2} + \frac{b^2}{a^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} + \frac{c^2}{b^2} \ge \frac ab + \frac ac + \frac ba + \frac bc + \frac ca + \frac ca$$
Now note t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove $\sum\limits_{i =1}^{26} \frac{a_i}{\sum_{j =0}^{i} a_j^2} \leq \sqrt{26}$ Given arbitrary real numbers $a_i$,
Prove that
$$\sum_{i =1}^{26} \frac{a_i}{\sum_{j =0}^{i} a_j^2} \leq \sqrt{26}$$
where $a_0 = 1$
So it will look like:
$$\frac{a_1}{(1+a_{1}^2)} + \frac{a_2}{(1+ a_{1}^2 + a_{2}^2)} + \cdots + \fr... | Here is a proof for the more general inequality
$$
\sum_{i=1}^n \frac{a_i}{1 + \sum_{j=1}^i a_j^2} \leq \sqrt{n}.
$$
where $a_1, \ldots, a_n$ range over $\mathbb{R}$. The method is a little cumbersome so I would be interested in a more direct proof.
I assume this problem comes from a math competition and would be int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
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.