Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
how to show $\frac{1+\cos x+\sin x}{1+\cos x-\sin x} = \frac{1+\sin x}{\cos x}$ I know that $\frac{1+\cos x+\sin x}{1+\cos x-\sin x} = \frac{1+\sin x}{\cos x}$ is correct. But having some hard time proofing it using trig relations. Some of the relations I used are
$$
\sin x \cos x= \frac{1}{2} \sin(2 x)\\
\sin^2 x + \... | $$\frac{1 + \cos x + \sin x}{1 + \cos x - \sin x}$$
$$= \frac{(1 + \cos x + \sin x)(1 + \cos x + \sin x)}{(1 + \cos x - \sin x)(1 + \cos x + \sin x)}$$
$$= \frac{1 + \cos^2 x + \sin^2 x + 2\cos x + 2\sin x + 2\sin x \cos x}{1 + 2\cos x + \cos^2 x - \sin^2 x}$$
$$= \frac{2(1 + \cos x + \sin x + \sin x \cos x)}{1 + 2\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1185058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find all possible values of $ a^3 + b^3$ if $a^2+b^2=ab=4$. Find all possible values of $a^3 + b^3$ if $a^2+b^2=ab=4$.
From $a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(4-4)=(a+b)0$. Then we know $a^3+b^3=0$.
If $a=b=0$, it is conflict with $a^2+b^2=ab=4$.
If $a\neq0$ and $b\neq0$, then $a$ and $b$ should be one positive and one ... | To give a different perspective, I can't find anywhere in the problem statement which states $a$ and $b$ must be real. Therefore, assume $a,b \in \mathbb{C}$. Then the equation $a^3+b^3=0$ along with the condition $ab=4$ implies, by multiplying both sides by either $a^3$ or $b^3$ that $a^6+64 = 0$ and $b^6+64=0$. There... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
If $\gcd(a,b)=1$ then $\gcd(a^2+b^2,a+2ab)=1$ or $5$ The question is already in the title.
Show that if $\gcd(a,b)=1$ then $\gcd(a^2+b^2,a+2ab)=1$ or $5$.
I saw yesterday this exercise in a book and I tried many things but I managed to show only that if some prime $p$ ($p$ must be $\equiv1\pmod{4}$)
divides $a^2... | suppose $gcd(a,b)=1$ and let $d=gcd(a^2+b^2,a+2b)$
the identity
$(a^2+b^2)+(a+2b)(2b-a)=5b^2$
shows that $d$ divides $5b^2$ and
since $5a^2=5(a^2+b^2)-5b^2$ , $d$ divides $5a^2$
since $a$ and $b$ are relatively prime, $d$ divides $5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Find the derivative using the chain rule and the quotient rule
$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$.
Here is my work:
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4x^4\left(x+1\right)^3}{\left(x+1\right)^8}$$
I know the fin... | You calculation of $f'(x)$ is correct, though it can be simplified. To do this, we factor out $4x^3(x+1)^3\require{cancel}$ in the numerator.
$$\begin{align}f'(x) &= \frac{4x^3(x+1)^4-4x^4(x+1)^3}{(x+1)^8}\\\\
&= \frac{\color{blue}{4x^3(x+1)^3}\Big((x+1) - x\Big)}{(x+1)^8}\\\\
&= \frac {(4x^3\cancel{(x+1)^3})(1)}{\canc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
Solve $\int\frac{x}{\sqrt{x^2-6x}}dx$ I need to solve the following integral
$$\int\frac{x}{\sqrt{x^2-6x}}dx$$
I started by completing the square,
$$x^2-6x=(x-3)^2-9$$
Then I defined the substitution variables..
$$(x-3)^2=9\sec^2\theta$$
$$(x-3)=3\sec\theta$$
$$dx=3\sec\theta\tan\theta$$
$$\theta=arcsec(\frac{x-3}{3})$... | $$
\begin{aligned}
\int \frac{x}{\sqrt{x^2-6 x}} d x & \int \frac{x}{x-3} d\left(\sqrt{x^2-6 x}\right) \\
= & \int\left(1+\frac{3}{x-3}\right)\left(\sqrt{x^2-6 x}\right) \\
= & \sqrt{x^2-6 x}+3 \int \frac{d\left(\sqrt{x^2-6 x}\right)}{\sqrt{\left(\sqrt{x^2-6 x}\right)^2+9}} \\
= & \sqrt{x^2-6 x}+3 \sinh^{-1} \left(\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How to solve $\int\sqrt{1+x\sqrt{x^2+2}}dx$ I need to solve
$$\int\sqrt{1+x\sqrt{x^2+2}}dx$$
I've chosen the substitution variables
$$u=\sqrt{x^2+2}$$
$$du=\frac{x}{\sqrt{x^2+2}}$$
However, I am completly stuck at
$$\int\sqrt{1+xu} dx$$
Which let me believe I've chosen wrong substitution variables.
I've then tried lett... | Substitute $x=\frac{2-t}{2\sqrt t}$. Then, $dx=-\frac {2+t}{4t\sqrt t}\>dt$ and
\begin{align}
&\int\sqrt{1+x\sqrt{x^2+2}}\>dx \\
= & -\frac18 \int \frac{t+2}{t^2} \sqrt{ 4+4t-t^2}\>dt
= -\frac18 \int \frac{\sqrt{ 4+4t-t^2}}{t-2}d\left( \frac{(t-2)^2}t\right)\\
=& -\frac{t-2}{8t }\sqrt{ 4+4t-t^2}- \int \frac1{t\sqrt{ 4+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Integration of fraction with square root I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fraction... | Since $x^2 + 7$ shows up twice it would make a good first substitution. If $u = x^2 + 7$ then
$$\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}} \, dx = \frac 12 \int \frac{1}{u + \sqrt{u}} \, du.$$
Can you take it from there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1192434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Given $n\equiv 1\pmod 8$, show that the number of subsets of a $n$-element set, whose size is $0\pmod 4$ is $2^{n-2}+2^ \frac{n-3}{2}$ Given $n\equiv 1\pmod 8$, show that the number of subsets of a $n$-element set, whose size is $0\pmod 4$ is $2^{n-2}+2^ \frac{n-3}{2}$
I don't get this question. If I have an n-element ... | What you want is
$$\binom{n}{0}+\binom{n}{4}+\binom{n}{8}+\cdots$$
for $n\equiv 1\pmod 8$.
Note that we have the followings :
$$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}+\binom{n}{4}+\cdots+\binom{n}{n}=2^n$$
$$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\binom{n}{4}-\cdots+(-1)^n\binom{n}{n}=0$$
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$? I'm trying to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$ by using the formula:
$$ \int udv = uv -\int vdu $$
I supposed that $u=x$ s.t $du=dx$, and also that $dv=\frac{x}{(x^2+1)^2}dx$, but I couldn't calculate the last integral. what is the tick here?
the answer m... | $$\int\dfrac{x^2}{(x^2+1)^2}dx$$
$$=x\int\dfrac x{(x^2+1)^2}dx-\int\left[\frac{dx}{dx}\cdot\int\dfrac x{(x^2+1)^2}dx\right]dx$$
$$=x\cdot\frac{-1}{1+x^2}+\int\dfrac{dx}{1+x^2}=\cdots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Continued Fraction Algorithm for 113/50 The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me determine the numbers $a_k$ for $\frac{113}{50}$?
What I have Done:
$\frac{113}{50} =... | $$\begin{gathered}
\frac{{113}}{{50}} = \frac{{100}}{{50}} + \frac{{13}}{{50}} = 2 + \frac{1}{{\frac{{39}}{{13}} + \frac{{11}}{{13}}}} = 2 + \frac{1}{{3 + \frac{{11}}{{13}}}} \hfill \\
= 2 + \frac{1}{{3 + \frac{{11}}{{11 + 2}}}} = 2 + \frac{1}{{3 + \frac{1}{{1 + \frac{2}{{10 + 1}}}}}} \hfill \\
= 2 + \frac{1}{{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the Inverse Laplace Transforms Find the inverse Laplace transform of:
$$\frac{3s+5}{s(s^2+9)}$$
Workings:
$\frac{3s+5}{s(s^2+9)}$
$= \frac{3s}{s(s^2+9} + \frac{5}{s(s^2+9)}$
$ = \frac{3}{s^2+9} + \frac{5}{s}\frac{1}{s^2+9}$
$ = \sin(3t) + \frac{5}{s}\frac{1}{s^2+9}$
Now I'm not to sure on what to do.
Any help will... | $\frac{3s+5}{s(s^2+9)} = -\frac{5}{9} \frac{s}{s^2+9} + \frac{2}{3}\frac{3}{s^2+9}+\frac{5}{9}\frac{1}{s}$ with ILT $-\frac{5}{9}\cos(3t)+\frac{2}{3}\sin(3t)+\frac{5}{9}step(t)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
(probability) Stuck on a proof in Grinstead and Snell This is probably (no pun intended) very simple, but it has fooled me. On page 294 of Grinstead and Snell's book on probability they derive the pdf of X+Y where both X and Y have the standard normal distribution. The step in the proof that stops me is from: $$\frac{1... | Suppose $X,Y \sim N(0,1)$ and $Z=X+Y$, and let the corresponding densities be $f_X,f_Y,f_Z$. Then
$$
\begin{align*}
f_Z(z) &= \int_{-\infty}^\infty f_X(z-y) f_Y(y) \, dy \\ &=
\int_{-\infty}^\infty \frac{1}{2\pi} \exp \left(-\frac{(z-y)^2}{2}\right) \exp \left(-\frac{y^2}{2}\right) \, dy \\ &=
\frac{1}{2\pi} \exp \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1204162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to calculate the area covered by any spherical rectangle? Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a spherical surface with a radius $R$? i.e. How to find the area $A$ of rectangle in terms of length $l$, widt... | On a sphere of radius $R > 0$, a geodesic triangle with interior angles $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$ has area $R^{2}(\theta_{1} + \theta_{2} + \theta_{3} - \pi)$. One way you might proceed, therefore, it to triangulate your geodesic polygon (whatever its actual shape) and sum the areas of its triangular... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1205927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 0
} |
How to prove through induction How can I prove by induction that
$$\binom{2n}n<4^n\;?$$
I have solved for the base case, $n=1$, and have formulated the induction hypothesis. I was thinking about Pascal's identity for the rest, but have not been able to come up with a way to use it.
| First, show that this is true for $n=1$:
$\binom{2}{1}<4^{1}$
Second, assume that this is true for $n$:
$\binom{2n}{n}<4^{n}$
Third, prove that this is true for $n+1$:
$\binom{2n+2}{n+1}=$
$\frac{(2n+2)!}{(n+1)!\cdot(n+1)!}=$
$\frac{(2n)!\cdot(2n+1)\cdot(2n+2)}{(n)!\cdot(n+1)\cdot(n)!\cdot(n+1)}=$
$\frac{(2n)!\cdot(2n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
an operator question I know how the derivative operator $\Big(\frac{d}{dx}\Big)^n$ works. But then how does it work if I have $$\exp{\Big(a\frac{d}{dx}+b\frac{d^2}{dx^2}\Big)}f(x)$$ I thought to use $$\exp z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$$ to have
\begin{align}
\exp{\Big(a\frac{d}{dx}+b\frac{d^2}{dx^2}\Big)... | For the operators $a \partial_{x} + b \partial_{x}^{2}$ it is seen that
\begin{align}
(a \partial_{x} + b \partial_{x}^{2})^{1} &= a \partial_{x} + b \partial_{x}^{2} \\
(a \partial_{x} + b \partial_{x}^{2})^{2} &= (a \partial_{x} + b \partial_{x}^{2})(a \partial_{x} + b \partial_{x}^{2}) \\
&= a^{2} \partial_{x}^{2} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find $p$ for which all solutions of system/equation are real There is system of $5$ equations
$$
a+b+c+d+e = p; \\
a^2+b^2+c^2+d^2+e^2 = p; \\
a^3+b^3+c^3+d^3+e^3 = p; \\
a^4+b^4+c^4+d^4+e^4 = p; \\
a^5+b^5+c^5+d^5+e^5 = p, \\
\tag{1}
$$
where $p\in\mathbb{R}$.
One can prove (like here) that $a,b,c,d,e$ are roots of eq... | Using the mean value theorem repeatedly (by taking 3 derivatives), we get that $$60x^2-4 \dbinom{p}{1}x+6 \dbinom{p}{2}$$ has to have $2$ real roots. This only happens if $$\Delta_x > 0 \iff 0 < p < \frac{60}{59}.$$
Using Geogebra, it seems that none of the values in this range produce a function with $5$ real roots so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Predicting increasing income within a given time I have this game where player earns "gold" and "reputation". With higher reputation, the player nets more income (gold).
*
*Each reputation points yields 1% bonus in income (gold)
*For every 20 gold received, player receives 1 reputation points
*Player starts with r... | Let $G(n)$ and $R(n)$ denote the amounts of gold and reputation after $n$ seconds, respectively. Then $G(0)=R(0)=0$ and $R(n)=\lfloor\tfrac{G(n)}{20}\rfloor$ and
$$G(n+1)=G(n)+10(1+0.1R(n))=G(n)+10(1+0.01\lfloor\tfrac{G(n)}{20}\rfloor).$$
If we ignore the floor function, this becomes $G(n+1)=1.005G(n)+10$, which can b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does this series $\sum_{i=0}^n \frac{4}{3^n}$ diverge or converge? I a newbie to series, and I have not done too much yet. I have an exercise where I have basically to say if some series are convergent or divergent. If convergent, determine (and prove) the sum of the series.
This is the first series:
$$\sum_{i=0}^n \fr... | Here is a simple proof for the geometric convergence:
Suppose we have the finite sum:
$$1+r+r^2+r^3+r^4+\dots+r^k$$ with $r\neq 1$
Since this is a finite sum, it will converge to some unknown quantity: $S$
So:
$$\begin{array}{lrl} S & =& 1+r+r^2+r^3+r^4+\dots+r^k\\
r\cdot S & =& ~~~~~~~ r+r^2+r^3+r^4+r^5+\dots+r^{k+1}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1210267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Find the derivative of the function $F(x) = \int_{\tan{x}}^{x^2} \frac{1}{\sqrt{2+t^4}}\,dt$. $$\begin{align}
\left(\int_{\tan{x}}^{x^2} \frac{1}{\sqrt{2+t^4}}\,dt\right)' &= \frac{1}{\sqrt{2+t^4}}2x - \frac{1}{\sqrt{2+t^4}}\sec^2{x} \\
&= \frac{2x}{\sqrt{2+t^4}} - \frac{\sec^2{x}}{\sqrt{2+t^4}} \\
&= \frac{2x-\sec^2{x... | $\text{ Recall } \int_{a(x)}^{b(x)}f(t) dt=F(b(x))-F(a(x)) \text{ *note:} \text{( where } F'=f) \\ \frac{d}{dx}[F(b(x))-F(a(x))]=\frac{d}{dx}F(b(x))-\frac{d}{dx}F(a(x)) \text{ by difference rule } \\ \\ \text{ now you use chain rule and you are almost there } $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Proving $(n+1)!>2^{n+3}$ for all $n\geq 5$ by induction I am stuck writing the body a PMI I have been working on for quite some time.
Theorem: $∀n∈N ≥ X$, $(n+1)!>2^{n+3}$
I will first verify that the hypothesis is true for at least one value of $n∈N$.
Consider $n=3$: (not valid)
$$(3+1)!>2^{3+3} \implies 4!>2^{6} \imp... | For $n\geq 5$, let $S(n)$ denote the statement
$$
S(n) : (n+1)! > 2^{n+3}.
$$
Base step $(n=5)$: $S(5)$ says $6!>2^8$, and this is true since $720>256$.
Inductive step: Fix some $k\geq 5$ and suppose that
$$
S(k) : (k+1)!>2^{k+3}
$$
is true. We must now prove that $S(k+1)$ follows where
$$
S(k+1) : (k+2)! > 2^{k+4}.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1214914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$ Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$
This is what I have so far:
Base case: f... | Hint: rewrite the sum as
$$
1-\left(\frac12-\frac13\right)-\left(\frac14-\frac15\right)-\left(\frac16-\frac17\right)-\dots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1216484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the least positive integer with $24$ positive divisors.
Find the least positive integer with $24$ positive divisors.
My attempt:
$24=2^3.3$. We shall have to find out a positive integer (least) $n$ such that $N$ has $24$ positive divisors i.e we have to find $N$ where $\tau(N)=24.$.
We have, if $N=p_{1}^{\alpha_... | We use the fact $p_1^{\alpha_1}p_2^{\alpha_2}\dots p_n^{\alpha_n}$ has $(\alpha_1+1)(\alpha_2+1)\dots(\alpha_n+1)$ divisors, and check minimum integer for each factorization, the possible factorizations for $24=2\cdot2\cdot2\cdot3$ are the following:
$2\cdot2\cdot2\cdot3$ minimum for this factorization is $2^2\cdot3\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1217233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
limit of $\frac{xy-2y}{x^2+y^2-4x+4}$ as $(x,y)$ tends to $(2,0)$ Am I able to substitute $x$ by ($k$+$2$) with $k$ tending to $0$, then using polar coordinates to deduce its limit?!
\begin{equation*}
\lim_{(x,y)\to(2,
0)}{xy-2y\over x^2+y^2-4x+4}.
\end{equation*}
| Take polar coorinates: $$x=2+r \cos{\theta}$$ $$y=r \sin{\theta}$$
$r \rightarrow 0 \Rightarrow x \rightarrow 2$
Thus $$\lim_{r \rightarrow 0}f(r, \theta)=\lim_{r \rightarrow 0} \frac{(2+r\cos{\theta}-2)(r \sin{\theta})}{(2+r\cos{\theta})^2+r^2 \sin^2{\theta}-4(2+r\cos{\theta})+4}= \lim_{r \rightarrow 0} \frac{r^2 \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1218520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Integration By Parts (Logarithm) $$\int(2x+3)\ln (x)dx$$
My attempts,
$$=\int(2x\ln (x)+3\ln (x))dx$$
$$=2\int x\ln (x)dx+3\int \ln(x)dx$$
For $x\ln (x)$, integrate by parts,then I got
$$=x^2\ln (x)-\int (x) dx+3\int \ln(x)dx$$
$$=x^2\ln (x)-\frac{x^2}{2}+3\int \ln(x)dx$$
For $\ln(x)$, integrate by parts, then I got
... | Bookish answer is wrong:
$$\left(x^2\ln (x)-\frac{x^2}{2}+\frac{3}{x}+c\right)'=2x\log x-\frac3{x^2}$$
And yours:
$$\left(\frac{1}{2}x(-x+2(x+3)\ln (x)-6)+c\right)'=2x\log x+3\log x=(2x+3)\log x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction Question:
Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case.
Attempt at solution:
So I plugged in $... | Your base case is wrong. You should realize it's true since $\frac{5}{4}<\frac{3}{2}$ obtained from $$1+\frac{1}{2^2} = \frac{5}{4} < \frac{3}{2} = 2-\frac{1}{2}$$ For the induction step, suppose $P(n)$ is true for all $n \in \{1,2,\ldots,k\}$. Then $$\begin{align}1+\frac{1}{2^2}+\ldots+\frac{1}{k^2}+\frac{1}{(k+1)^2} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1220203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Why is the sum of the digits in a multiple of 9 also a multiple of 9? The sum of the digits in $9 k$ (where $k$ is an integer) is a multiple of $9$: for example
$$9\cdot 1=9$$
$$9\cdot 7=63 \qquad \text{and } 6+3=9\cdot 1$$
$$9\cdot 11=99 \qquad \text{and } 9+9=9\cdot 2$$
But why?
| It is because $10$ gives $1$ as remainder when dividing by $9$. This can be expressed by the general notation of 'congruence'
$$10\equiv 1\pmod9$$
(that expresses that these two numbers give the same remainder when dividing by $9$).
Then, it easily follows from the property of these congruences that they behave like e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
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A bessel function integral $$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$
How do I show this?
| This is going to be a little involved. Begin with the representation
$$\int_{-1}^1 du \frac{u}{\sqrt{1-u^2}} \sin{r u} = \pi J_1(r) $$
which may be obtained by differentiating the well-known relation
$$\int_{-1}^1 du \frac{\cos{r u}}{\sqrt{1-u^2}} = \pi J_0(r) $$
Then
$$\begin{align}\int_{-\infty}^{\infty} dy \frac{J_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1221767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
$ax^2+bx+c=0$ has roots $x_1,x_2$. what are the roots of $cx^2+bx+a=0$. Given solution: Dividing the first equation by $x^2$ we get
$c(\frac{1}{x^2})+b(\frac{1}{x})+a=0$
so $(\frac{1}{x_1}),(\frac{1}{x_2})$ are the roots of $cx^2+bx+a=0$.{How?It is not obvious to me.}
The answers so far are proving retrospectively that... | The definition says: $t$ is a root of a polynomial $P$ if and only if $P(t)=0$.
so let $P(x)=cx^2+bx+a$, we have :
$$P(\frac{1}{x_1})=c(\frac{1}{x_1^2})+b(\frac{1}{x_1})+a=\frac{c+bx_1+ax_1^2}{(x_1)^2}$$
and as we know $x_1$ is a root of $ax^2+bx+c=0$ hence $ax_1^2+bx_1+c=0$ and finally:
$P(\frac{1}{x_1})=0$
as a concl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Prove $10n^8 - 9n^6 - n^2$ is divisible by $45$ Basically, I have to use Euler theorem to prove that
$10n^8 -9n^6 -n^2$ is divisible by $45$
So my approach so far is to say
$10n^8 - 9n^6 - n^2 = 0 \bmod 45$
Now $45$ can be factored into $5$ and $9$
So I figure I need to do something like
$10n^8 - 9n^6 - n^2 \bmod 5$
$... | HINT
$$10n^8 - 9n^6 - n^2\equiv 0n^8-(-1)n^6 - n^2 \equiv n^6 - n^2 \pmod 5$$
By little fermat we have $n^5\equiv n \pmod 5$, therefore
$$n^6-n^2\equiv n\cdot n^5-n^2\equiv n\cdot n-n^2\equiv n^2-n^2\equiv 0\pmod{5}$$
See if you can work the other mod similarly
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1225018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to solve $\int_0^{\frac{\pi}{2}}\frac{x^2\cdot\log\sin x}{\sin^2 x}dx$ using a very cute way? Few days ago my friend gave me this integral and i cant get how to solve this. The integral is:$$\int_0^{\large \frac{\pi}{2}}\frac{x^2\cdot\log{{\sin{x}}}}{\sin^2{x}}dx$$
| Hint. You may observe that
$$
\frac{\log{{\sin{x}}}}{\sin^2{x}}=\left( -\frac{x^2}{2}-\log \sin x-\frac{1}{2} \log^2 \sin x\right)''
$$ thus integrating by parts twice, using $(x^2)''=2$, leads to
$$
\begin{align}
\int_0^{\pi/2}\frac{x^2\cdot\log{{\sin{x}}}}{\sin^2{x}}dx&=2\int_0^{\pi/2}\left( -\frac{x^2}{2}-\log \sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
quadratic equation what am I doing wrong? solve
$$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} $$
$$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} \Rightarrow $$
$$ 5x+19 = (x+7) + 4\sqrt{x-5}\sqrt{x+7} + (x+5) \Rightarrow $$
$$ 3x + 17 = 4\sqrt{x-5}\sqrt{x+7} \Rightarrow $$
$$ 9x^2 + 102x + 289 = 16(x+7)(x-5) \Rightarrow $$
$... | You forgot a factor $4$ and wrote $x+5$ instead of $x-5$ in the third line, which should be
$$
5x+19=x+7+4\sqrt{x+7}\,\sqrt{x-5}+4(x-5)
$$
giving
$$
4\sqrt{x+7}\,\sqrt{x-5}=32
$$
or
$$
\sqrt{x+7}\,\sqrt{x-5}=8
$$
that becomes, after squaring,
$$
x^2+2x-99=0
$$
The roots of this are $-11$ and $9$, but only the latter is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Limit of $x^2e^x $as $x$ approaches negative infinity without using L'hopital's rule I'm trying to find the limit of $e^x x^2$ as $x$ approaches negative infinity:
$$\lim_{x\to-\infty}e^xx^2$$
without using L'hopital's rule. Any suggestions?
| $$\lim_{x\rightarrow -\infty} x^2e^x\\y=-x\\so \\\lim_{y\rightarrow +\infty} (-y)^2e^{(-y)}=\\\lim_{y\rightarrow +\infty} \frac{y^2}{e^y}=?$$ as you know :$e^y=1+y+\frac{y^2}{2!}+\frac{y^3}{3!}+...$ now see $$\lim_{y\rightarrow +\infty} \frac{y^2}{e^y}= \\lim_{y\rightarrow +\infty} \frac{y^2}{1+y+\frac{y^2}{2!}+\frac{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Polynomials mod prime $p$ The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do this. Then, I am supposed to use the process of completing the square to convert it to a congrue... | Multiply by the inverse of $5$, which is $3$:
$$ 3 \times ( 5 m^2+m+4) \equiv 0 \pmod{7}, $$
which becomes
$$ m^2+3m+5 \equiv 0 \pmod{7} $$
Completing the square means finding $a$ and $b$ so the following is true
$$ m^2+3m+5 \equiv (m+a)^2+b \pmod{7} $$
Expanding the bracket,
$$ 3m+5 \equiv 2am + (a^2+b) \pmod{7} $$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ It is given that $4 x^4 + 9 y^4 = 64$.
Then what will be the maximum value of $x^2 + y^2$?
I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .
| One more stab at the problem. Instead of working with $x$, let's work with $u = x^2$ as it makes the algebra slightly easier.
As $4x^4 + 9y^4 = 64$, we have $y^2 = \frac{1}{3}\sqrt{64-4u^2}$, taking only the positive square roots as $y^2 \geq 0$. Hence want to maximize
$$f(u) = x^2 + y^2 = u + y^2 = u + \frac{1}{3}\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Sufficient conditions to guarantee a series is Cesaro summable Is is true that every nonnegative, bounded series in $R$ is Cesaro summable?
Is there a list of sufficient conditions on series to guarantee that it is Cesaro summable?
| In the other case, if you mean the partial sums are non-negative and bounded, no, it is not true that every non-negative bounded series in R is Cesaro summable! This is from somewhere in Hardy's brilliant book 'Divergent Series', look for 'dilution'.
take $a_n = (-1)^n$ and $c_n = \begin{cases}a_k & n=2^k \\ 0 & \text{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all possible two-way associations/relations between four numbers Given four numbers {1,2,3,4}, how to find all possible two-way associations/relations between them? I calculate them manually as in below (50 in total) but I would like to know whether a mathematical formula exists to find them? And how about genera... | Create the set $A$ by choosing at least $1$ element from $\{1, 2, 3, ..., n\}$ and leaving at least one for $B$. Suppose we chose $k$ elements ($1\leq k\leq n-1$). Now, from the remaining elements, we need to choose elements for $B$. There are $n-k\geq 1$ elements left and we must choose at least one of them. Say w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Use the generating function to solve a recurrence relation We have the recurrence relation $\displaystyle a_n = a_{n-1} + 2(n-1)$ for $n \geq 2$, with $a_1 = 2$.
Now I have to show that $\displaystyle a_n = n^2 - n +2$, with $n \geq 1$ using the generating function.
The theory in my book is scanty, so with the help o... | We have $a_n - a_{n-1} - 2n +2 = 0 \ (\star)$. Suppose the GF of $\langle a_n \rangle_{n\ge 1}$ is $f(x)$.
Then,
$$\begin{align*}
f(x) &= a_1 + &a_2 x& + a_3 x^2 + \cdots + a_n x^n + \cdots \\
-xf(x) &= &-a_1 x& - a_2 x^2 - \cdots - a_{n-1}x^n - \cdots\\
\frac{-2x}{(1-x)^2} &= &-2 x& - 4 x^2 \ \ - \cdots - 2n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$. I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$
where $B_n$ is the $n$th Bell number.
I found this result by exploring the series on wolframalpha and looking up the sequence of numbers generated. I have no experienc... | Notice that
\begin{align}
S
& = \sum_{n = 0}^{\infty} \frac{n^{2}}{n!} \\
& = \frac{0^{2}}{0!} + \frac{1^{2}}{1!} + \frac{2^{2}}{2!} + \frac{3^{2}}{3!} +
\cdots \\
& = \frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + \frac{4}{3!} + \cdots.
\end{align}
Then
\begin{align}
S - e
& = \left(
\frac{1}{0!} + \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Differentiate $(x + 1)(x + 2)^2(x + 3)^3$ Obviously we can "brute force" this by multiplying the various terms and differentiating from there. But based upon the solution provided in the text where I found this problem, it looks like there's a "cleaner" method available.
Of note, this problem appears in the chapter int... | There is also the logarithmic differentiation approach: http://en.wikipedia.org/wiki/Logarithmic_differentiation
Note that $\ln((x+1)(x+2)^2(x+3)^3)=\ln(x+1)+2\ln(x+2)+3\ln(x+3)$.
Then $\frac{d}{dx}\ln(f(x))=\frac{f^{\prime}(x)}{f(x)}$ using the chain rule.
So take the derivative of the RHS (above) to get
$$\frac{f^\pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Contest Problem Let N be the set of positive integers whose decimal representation
is of the form 4ab4 (for example: 4174 or 4004). If a number x is
picked randomly from N, what is the probability that x is divisible
by 6, but not divisible by 12?
I got an answer of $\frac{17}{100}$. Could you someone verify that for ... | $N$ has $100$ members.
Every member $x$ of $N$ is divisible by $2$; it is divisible by $3$ (and therefore by $6$) iff $a+b \equiv 1 \mod 3$. Of these, $x$ is divisible by $12$ iff $b$ is even. There are three cases $(b = 3, 5, 9$) with $b$ odd and three possible $a$'s ($a = 1,4, 7$ for $b = 3$ or $9$, $a = 2,5,8$ fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the solution to this binomial? what's the coefficient of $x^6$ in the expansion of $(1+X^2+X)^{-3}$?
I have factorized the term to $\left(\frac{1-x^{-3}}{1-x}\right)^3$
after this I'm having problem solving it
| According to Wolfram alpha, the Taylor series of your function is
$$1-3 x+3 x^2+2 x^3-9 x^4+9 x^5+3 x^6-18 x^7+18 x^8+4 x^9-30 x^{10}+\dots.$$
You can calculate this by solving linear equations. For example, here is how to find the first three coefficients. Let the beginning of the Taylor expansion be $A + Bx + Cx^2 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convert Function into Spherical Coordinates $$z=\sqrt{4-x^2-2y^2} $$
First thing I did was put the equation in standard form: $z^2+x^2+2y^2=4 $
Then I convert to spherical: $$\rho^2\cos^2(\phi)+\rho^2\sin^2(\phi)\cos^2(\theta)+2\rho^2\sin^2(\phi)\sin^2(\theta)=4$$
then I simplify:
$$\rho^2[\cos^2(\phi)+\sin^2(\phi)\co... | There is a mistake in your last line, it should be
$$\rho^2[\cos^2\phi+\sin^2\phi\left[\cos^2\theta+2\sin^2\theta]\right]=4$$
so that
$$\rho^2[\cos^2\phi+\sin^2\phi\left(1+\sin^2\theta)\right]=4$$
$$\rho^2(\cos^2\phi+\sin^2\phi+\sin^2\phi\sin^2\theta)=4$$
$$\rho^2\left((\cos^2\phi+\sin^2\phi)+\sin^2\phi\sin^2\theta\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many distinct possible forms for its Jordan canonical matrix are there? 4x4 non-diagonalizable matrix with two unique eigenvalues I know the sum of $A_m$ equals $4$ as $\dim(A) = 4$ and sum of $G_m$ can't equal $4$ as $A$ is non-diagonalizable.
After I write down all the cases, what should I do?
| After you write down the cases, count how many you have. I count 24. Following is my work where the first entry in an ordered pair refers to the first eigenvalue, and the second to the second distinct eigenvalue. Note that without a convention regarding Jordan canonical form, blocks can be permuted.
\begin{array}{ccc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Determining the Laurent Series I need to determine the Laurent series of this function:
$$\frac{1}{(z-1)(z+5)}$$ Inside the annulus: $$\left\{z|1<|z-2|<6\right\}$$ Any help appreciated.
| Using partial fraction expansion, we can write
$$\frac{1}{(z-1)(z+5)}=\frac{1/6}{z-1}-\frac{1/6}{z+5}$$
We seek a series about the center of the annulus $z=2$. To that end,
$$\begin{align}
-\frac{1/6}{z+5}&=\frac16 \frac{1}{(z-2)+7}\\\\
&=-\frac{1}{42}\frac{1}{1+\frac{z-1}{7}}\\\\
&=-\frac{1}{42} \sum_{n=0}^{\infty} (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $\sqrt{x^2} = (\sqrt x)^2$? Take $x=4$ for example:
$ \sqrt{(4)^2} = \sqrt{16} = \pm4 $
However:
$ (\sqrt{4})^2 = \sqrt{\pm2}$
Case 1: $ (-2)^2 = 4$
Case 2: $ (2)^2 = 4$
Solution : $+4$
How come the $ \sqrt{(4)^2} = \pm4$; but $ (\sqrt{4})^2 = 4 $ ?
What is missing?
| $\sqrt{x^2} = \sqrt{x \cdot x} = \sqrt{x} \cdot \sqrt{x} = \sqrt{x}^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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How to Find the Function of a Given Power Series? (Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.)
Can someone please explain how to find the power series expansion of the function in the title? Also, how would you... | To answer both your old and your new question at the very same time, we can consider a surprising relationship between power series and recurrence relations through generating functions. As a simple example, consider representing $\frac{1}{1-x}$ as a power series. In particular, we want to discover an $f_n$ such that
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
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Evaluate $\text{k}$ from the given equation
If
$$ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + \text{k} \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x =0$$
then find the value of $\text{k}$
My Approach :
Let
$\text{I}= \displaystyle \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\rig... | The crux to completing your work is to prove the following
$$\int_0^1 \frac{\log^2(1-x)}{x^2} \,\mathrm{d}x
= -2 \int_0^1 \frac{\log(1-x)}{x}\,\mathrm{d}x$$
You are close though, let me show you how to finish your work. I really liked this problem. Summarizing your work, you have proven that
$$
\int_0^\infty \left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$? Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$?
Source: calculus made easy by S. Thompson.
I have looked up the formula for binomial theorem with negative exponents but it is confusing. The expansion stated in the text is:
$$\left[1-\frac{2\,dx}{x}... | First, recall the expansion $\frac{1}{1-x}=1+x+x^2+\cdots$. If you don't believe this yet, let $z=1+x+x^2+\cdots$, then $(1-x)z=z-xz=(1+x+x^2+\cdots)-(x+x^2+x^3+\cdots)=1$, so $z=\frac{1}{1-x}$.
Next, we can derive an expansion of $\frac{1}{(1-x)^2}$ by squaring this sequence: $\frac{1}{(1-x)^2}=\left(1+x+x^2+\cdots\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1250593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find the solution to the differential equation Assume $x\gt 0$ and let
$$x(x+1)\frac{du}{dx} = u^2,$$
$$u(1) = 4.$$
I started off by doing some algebra to get:
$$\frac{1}{u^2}du = \frac{1}{x^2+x}dx.$$
I then took the partial fraction of the right side of the equation:
$$\frac{1}{u^2}du = \left(\frac{1}{x}-\frac{1}{x+1... | Solving for $\frac{du}{dx}$ we have
\begin{equation*}
\frac{du}{dx}=\frac{u^2}{x(x+1)}\\
\Rightarrow \frac{du}{dx}=\frac{u^2}{x^2+x}\\
\frac{\frac{du}{dx}}{u^2}=\frac{1}{x^2+x}.
\end{equation*}
Integrate both sides & evaluate the integrals:
\begin{equation*}
-\frac{1}{u}=\log(x)-\log(x+1)+C_1\\
\Rightarrow u=-\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving that $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{13}{24}$ by induction. Where am I going wrong? I have to prove that
$$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n}>\frac{13}{24}$$ for every positive integer $n$.
After I check the special cases $n=1,2$, I have to prove that the given inequality holds fo... | At first sight it seems to be false because 1/n tends to 0. But I trust you so your inequality is nice. Want I want to remark is you have at hand a pretty proof that the harmonic series is divergent (You have an infinity of "packages" each of them greater than 13/24)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Binomial sum identity Everyone knows that $\sum{n \choose 2k}=\sum{n \choose 2k+1}$
My question is follwing
What is the clear form of $\sum_{k=0} (-1)^k{n \choose 2k}$ and $\sum_{k=0} (-1)^k{n \choose 2k+1}$
For example, my exercise is
"calculate a+b where a=$\sum_{k=0}^{25} (-1)^k{50 \choose 2k}$ & b=$\sum_{k=0}^{24} ... | Start with
$$
(1 + i)^{n}
=
\sum_{k=0}^{n} \binom{n}{k} i^{k}
=
\sum_{h} \binom{n}{2 h} (-1)^{h}
+
i \sum_{h} \binom{n}{2 h + 1} (-1)^{h},
$$
and then note that
\begin{align}
(1 + i)^{n}
&=
\sqrt{2}^{n} \cdot \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right)^{n}
\\&=
\sqrt{2}^{n} \cdot \left( \cos\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding both max and min points by Lagrangian Using the method of the Lagrange multipliers, find the maximum and minimum values of the function
$$f(x,y,z) = x^2y^2z^2$$
where $(x,y,z)$ is on the sphere
$$x^2 +y^2 +z^2 = r^2$$
So $L(x,y,z;\lambda)=x^2y^2z^2 + \lambda (x^2 +y^2 +z^2-r^2)$.
After finding $L_{x,y,z,\lambda... | So the equations you find by setting the derivatives equal to zero are
$$ 2x( y^2 z^2 + \lambda) = 0 \\
2y (x^2 z^2 + \lambda) = 0 \\
2z (y^2 x^2 + \lambda) = 0 \\
x^2 + y^2 + z^2 - r^2 = 0
$$
Right, let's stop here and think. Suppose $x=0$. Then the first equation's obviously satisfied. The sphere equation can obviou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ ... Given two integer sequences
\begin{equation*}
\displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor ,
\end{equation*}
\begin{equation*}
B_n=\left\lfloor\dfrac{n^2}{2\varphi}\right\rfloor-\left\lfloor \dfrac{n}{2\varphi^2}\right\... | I can prove less; we have $$\frac {n^2 + n} {2 \varphi} - n < A_n \leqslant \frac {n^2 + n} {2 \varphi}$$ and $$\frac {n^2 - n/\varphi} {2 \varphi} - \frac {1} {2} < B_n \leqslant \frac {n^2 - n/\varphi} {2 \varphi}.$$ It follows that $$\frac {n (\varphi + 1 - 2 \varphi^2)} {2 \varphi^2} < A_n - B_n < \frac {(\varphi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1259142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$
Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$
I must use Vieta's formulas in my solution since that is the lesson we are covering with our teacher.
My solution:
Let p and q be the roots of the q... | Let $r=p+6$ and $s=q+6$ be the roots of the quadratic you're trying to find. That quadratic's coefficients are $-(r+s)$ and $rs$. But
$$\begin{align}
r+s&=(p+6)+(q+6)\\
&=p+q+12\\
&=-8+12\\
&=4
\end{align}$$
and
$$\begin{align}
rs&=(p+6)(q+6)\\
&=pq+6(p+q)+36\\
&=-1+6(-8)+36\\
&=-1-48+36\\
&=-13
\end{align}$$
so the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1259949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Limit without applying l'hopital's rule, $\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$. This is the question:
$$\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$$
I know the answer is $-1$, but can someone go through the steps and explaining it to me?
| You should better be aware of the definition of a modulus function.
$$|x| = \left\{\begin{matrix}
x & x > 0\\
0 & x = 0 \\
-x & x<0
\end{matrix}\right.$$
For this question,
$$|2x+5| = \left\{\begin{matrix}
2x+5 & 2x+5 > 0\\
0 & 2x+5 = 0 \\
-(2x+5) & 2x+5<0
\end{matrix}\right.$$
Here, since, $x \to -\infty$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
What is the value of $\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}$ If $\frac{a}{a^{2}+1} = \frac{1}{3}$ Then find the value of $$\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}$$
Any hints to help me?
| Since $a+\frac 1a=3$, one has
$$a^2+\frac{1}{a^2}=\left(a+\frac 1a\right)^2-2=7$$
$$a^3+\frac{1}{a^3}=\left(a+\frac 1a\right)^3-3\left(a+\frac 1a\right)=18.$$
Now note that
$$\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}=\frac{1}{a^3+\frac{1}{a^3}+a^2+\frac{1}{a^2}+a+\frac 1a+1}=\frac{1}{18+7+3+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
A possible dilogarithm identity? I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before?
$$2 \left(\text{Li}_2\left(2-\sqrt{2}\right)+\text{Li}_2\left(\frac{1}{2+\sqrt{2}}\right)\right)+\text{Li}_2\left(3-2 \sqrt{2}\right)$$
| Euler's reflection identity states:
$$\operatorname{Li}_{2}{\left(z\right)}+\operatorname{Li}_{2}{\left(1-z\right)}=\zeta{(2)}-\ln{\left(z\right)}\ln{\left(1-z\right)}.$$
Landen's dilogarithm identity states:
$$\operatorname{Li}_{2}{\left(z\right)}+\operatorname{Li}_{2}{\left(\frac{z}{z-1}\right)}=-\frac12\ln^2{\left(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Write 100 as the sum of two positive integers
Write $100$ as the sum of two positive integers, one of them being a multiple of $7$, while the other is a multiple of $11$.
Since $100$ is not a big number, I followed the straightforward reasoning of writing all multiples up to $100$ of either $11$ or $7$, and then find... | From Bezout's Lemma, note that since $\gcd(7,11) = 1$, which divides $100$, there exists $x,y \in \mathbb{Z}$ such that $7x+11y=100$.
A candidate solution is $(x,y) = (8,4)$.
The rest of the solution is given by $(x,y) = (8+11m,4-7m)$, where $m \in \mathbb{Z}$. Since we are looking for positive integers as solutions, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1265426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 8,
"answer_id": 0
} |
What wolfram does to factor $x^6+x^2+2$? I am learning polynomials and I am trying to understand what wolfram did to obtain $$(x^2+1)(x^4-x^2+2)$$ from $$x^6+x^2+2$$
It does not show me the step-by-step option in this case and I got confused. I only see $$x^2(x^4+1)+2$$ for this case.
Thank you.
| Notice that $x^6+x^2+2=(x^6+1) + (x^2+1)$.
Since $x^6+1$ is a sum of cubes it factors like this: $$x^6+1=(x^2+1)(x^4-x^2+1).$$ Putting these together gives $$x^6+x^2+2=(x^2+1)(x^4-x^2+1)+(x^2+1)=(x^2+1)(x^4-x^2+2).$$
Not sure about Wolfram's actual algorithm though. For algorithms see Wikipedia. For an explanation of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Evaluating a triple integral in spherical coordinates I need to evaluate the integral $\int \int \int \frac{x^2}{x^2+y^2}$ over the region $D$ where $D = {(x,y)} : 1\leq x^2+y^2+z^2 \leq 2, z^2>=x^2+y^2$ and $z\leq 0$
So I tried converting to spherical coordinates, therefore $1\leq r\leq\sqrt{2}$
and $0\leq \theta \leq... | $$
I = \int\limits_D \frac{x^2}{x^2+y^2}\, dV
$$
$D$: $1 \leq x^2+y^2+z^2 \leq 2$ (spherical shell between $r=1$ and $r=\sqrt{2}$), $z^2>=x^2+y^2$ and $z\le 0$ (negative cone).
Going to spherical coordinates $dV = r \sin\theta\,dr\,d\theta\,d\phi$, $x = r \sin \theta \cos \phi$, $y = r \sin \theta \sin \phi$ gives
\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof using partial fractions I have to prove this formula:
$$\int \! \frac{1}{(x^2+\beta ^2)^{k+1}} \, \mathrm{d}x=\frac{1}{2k\beta ^2}\frac{x}{(x^2 +\beta^2)^k}+\frac{2k-1}{2k\beta^2}\int \! \frac{1}{(x^2+\beta ^2)^k} \, \mathrm{d}x
$$
I have to use partial fraction decomposition.
I started like this:
$$
\frac{a}{(x^... | Write
$$\begin{align}
\frac{1}{(x^2+\beta^2)^{k+1}}&=\frac{1}{\beta^2}\frac{(x^2+\beta^2)-x^2}{(x^2+\beta^2)^{k+1}}\\\\
&=\frac{1}{\beta^2}\left(\frac{1}{(x^2+\beta^2)^{k}}- \frac{x^2}{(x^2+\beta^2)^{k+1}}\right)
\end{align}$$
Now, integrate both sides and use integration by parts on the last term. Combine terms of li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
To find jordan canonical form Which of the following matrices have Jordan canonical form of equal to the $3\times 3$ matrix
$$
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$$
a)$
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$
b)$
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 1 \\
0 & 0 ... | The given Jordan canonical form implies, minimal polynomial of corresponding matrix should be $x^2=0.$ Hence if matrix $A$ is having the property that $A \neq 0$ and $A^2=0 ,$ it will have desired Jordan canonical form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
transformation of a difference equation How can I translate the difference equation
$$x_{k+3}+4x_{k+2}+3x_{k+1}+x_k=2u_{k+2}$$
into a state-space representation of the following form (A and B are matrices)
$$x_{k+1}=Ax_k+Bu_k$$
| As always in such cases. Let
$$ X_k := \begin{pmatrix} x_k \\ x_{k+1}\\ x_{k+2} \end{pmatrix} $$
and
$$ U_k := \begin{pmatrix} u_k \\ u_{k+1} \\ u_{k+2}\end{pmatrix} $$
Then the above can be written as
\begin{align*}
X_{k+1} &= \begin{pmatrix} x_{k+1} \\ x_{k+2}\\ x_{k+3} \end{pmatrix} \\
&= \begin{pmatrix} 0 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$ Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$. I thought to write
$$\dfrac{2^2}{2^{\ln(x)}} + 2^2 \cdot 2^{\ln(x)} = 2^3 \implies \dfrac{2^2 + 2^2 \cdot \left(2^{\ln(x)}\right)^2}{2^{\ln(x)}} = 2^3$$
What should I do else?
| We have $2^{2-\ln(x)} = \dfrac4{2^{\ln(x)}}$ and $2^{2+\ln(x)} = 4\cdot2^{\ln(x)}$. Hence, setting $2^{\ln(x)} = a$, we obtain
$$\dfrac4a + 4a = 8 \implies a^2 + 1 =2a \implies a =1 \implies 2^{\ln(x)} = 1 \implies x = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
The limit of this function when x goes to minus infinity? I'm looking for the limit of $$\mathop {\lim }\limits_{x \to - \infty } \left( {\sqrt {1 + x + {x^2}} - \sqrt {1 - x + {x^2}} } \right)$$
I know it should be -1, but for some reason I always get to 1.
I'm not sure where the difference between $$- \infty$$ and ... | Here is another and perhaps quicker way to proceed. Use
$$\sqrt{1+t}=1+\frac12 t +O(t^{-2})$$
Here we have
$$\sqrt{x^2\pm x+1}=\sqrt{x^2(1\pm x^{-1}+x^{-2})}=|x|\left(1\pm \frac1{2x}+O\left(\frac{1}{x^2}\right)\right)$$
whereupon subtracting the "upper" and "lower" signed expressions yields
$$|x|\left(\frac{1}{x}+O\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$
Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$
Attempt:
I want to use the following theorem:
The largest exponent of $e$ of a prime $p$ such that $p^e$ is a divisor... | First in the définition of $e_1$ and $e_2$ there is no factoriel $!$ we have:
$$ e_1=\left\lfloor\frac{5^{25}-1}{5}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^2}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^3}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^4}\right\rfloor+\cdots+\left\lfloor\frac{{5^{25}-1}}{5^{24}}\right\rf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
$f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$ Let f be defined from real to real
$f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$
Now how to find the period of this function f(x)?
Can someone provide me a purely algebraic method to solve this problem please?
Update:... | For each fixed $y \in [0,1)$, your equation is a second order linear recurrence relation with constant coefficients. This suggests that the first step should be to solve the characteristic equation $\lambda^2-\sqrt{3} \lambda + 1 = 0$. You find $\lambda_1,\lambda_2 = \frac{\sqrt{3} \pm i}{2}=e^{\pm i \pi/6}$. This mean... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$ I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$
The equations are obviously equal, but some algebraic mani... | It's very easy. It comes by using factorization and simplification rules in general. In the case of your question, we have
$- \frac{1}{x-2} = \frac{(-1)}{(-1)(-x+2)}= \frac{(-1)}{(-1)}\times \frac{1}{(-x+2)}= \frac{1}{(-x+2)}$
and in the case of $\frac{1}{x-3}$, by multiplying both denominator and numerator with $(-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 2
} |
How to compute $\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$? I have to compute the series $\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$.
$$\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}} = \sum_{n=0}^{\infty}{\frac{3^n\frac{1}{2}}{n!}} + \sum_{n=0}^{\infty}{\frac{3^nn}... | As for integer $n>0, n!=n\cdot(n-1)!,$
$$\sum_{n=0}^{\infty}{\frac{3^nn}{n!}}=\sum_{n=1}^{\infty}{\frac{3^nn}{n!}}=3\sum_{n=1}^\infty\dfrac{3^{n-1}}{(n-1)!}=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1275769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Find $\int \frac{1}{x^4+x^2+1} \,\, dx$ Find $$\int \frac{1}{x^4+x^2+1} \,\, dx$$
I tried to find like that:
$\int \frac{1}{x^4+x^2+1} = \int \frac{\frac{1}{2}x + \frac{1}{2}}{x^2+x+1} \,\, dx + \int \frac{-\frac{1}{2}x + \frac{1}{2}}{x^2-x+1} \,\, dx = \frac{1}{2} \Big(\int \frac{2x + 1}{x^2+x+1} - \int \frac{x}{x^2+... | simple methods,Note
$$1=\dfrac{1}{2}(x^2+1)+\dfrac{1}{2}(1-x^2)$$
so
$$\int\dfrac{1}{x^4+x^2+1}dx=\frac{1}{2}(I_{1}+I_{2})$$
$$I_{1}=\int\dfrac{x^2+1}{x^4+x^2+1}dx=\int\dfrac{1+\frac{1}{x^2}}{x^2+\frac{1}{x^2}+1}dx=\int\dfrac{d(x-\frac{1}{x})}{(x-1/x)^2+3}$$
and simaler $I_{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Computing a limit similar to the exponential function I want to show the following limit:
$$
\lim_{n \to \infty}
n
\left[
\left( 1 - \frac{1}{n} \right)^{2n}
- \left( 1 - \frac{2}{n} \right)^{n}
\right]
= \frac{1}{e^{2}}.
$$
I got the answer using WolframAlpha, and it seems to be correct numerically, but I ... | $$
(1-\frac1{n})^{2n} =(1-\frac2{n}+\frac1{n^2})^n
$$
use the binomial expansion:
$$
(1-\frac2{n}+\frac1{n^2})^n =\sum_{k=0}^n(1-\frac2{n})^{n-k}\binom{n}{k}\frac1{n^{2k}}
$$
so
$$
n
\left[
\left( 1 - \frac{1}{n} \right)^{2n}
- \left( 1 - \frac{2}{n} \right)^{n}
\right] = \sum_{k=1}^n(1-\frac2{n})^{n-k}\binom{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Area of region inequality What is the area of the region defined by the following set of inequalities?
$$\begin{array}{cc} (1) &-1 < xy < 1 \\ (2) &-1 < x^2-y^2 < 1 \end{array}$$
I think we use integration here
| The area that we want is:
Define $u = x y$ and $v = x^2-y^2$. Then, $ {du} = y {dx} + x {dy}$ and $ {dv} = 2x {dx} - 2y {dy}$.
Solve this system of equations for $ {dx}$ and $ {dy}$ to get:
$ {dx} = \frac{y}{x^2+y^2} {du} + \frac{x}{2x^2+2y^2} {dv}$ and $ {dy} = \frac{x}{x^2+y^2} {du} - \frac{y}{2x^2+2y^2} {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How to solve for x in $2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}$ This is the question:
$$\large{2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}}$$
What I did was put $~\large{2^{x^{2}}=t}$
From this, I got, roots of the quadratic:
$$\large{-2^{x+1}\pm~\left( 2^{2x+1}\sqrt{3}\right)}$$
Now this is equal to $\boxed{\large{2^{x^2}}}$. How... | The given equation can be rewritten as $$2^{2x^2-4x+2}+2^{x^2-2x+4}=2^7$$Now take $2^{(x-1)^2}=z$ which gives you the equation $$z^2+8z=2^7\implies z=8,-16$$ Taking only the positive solution gives you, (assuming $x$ to be real)$$(x-1)^2=3\implies x=1\pm \sqrt{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cannot understand an Integral $$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } $$
I had to solve the integral and get it in this form.
My attempt:
$$\int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } $$
$$=\int _{\frac{\pi}{6}}^{ \frac{\pi}{3}} \dfrac{\sin x \cos x }{ \sin x+\cos ... | To evaluate the integral use the identity $$(\sin{x}+\cos{x})^2=1+2{\sin{x}}{\cos{x}}$$
And substitute $\sin(x)\cos(x) = u$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$
Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$
I don't have any olympic background, so I may be missing some standard trick.
The inequality looks closely related to the concavity of $\... | @math110 is right, but you have started with the wrong inequality. You can use concavity instead directly and get:
$$\frac{x}{x+y}\ln \left(\frac{a}x \right)+\frac{y}{x+y}\ln \left(\frac{b}y \right) \le \ln\left(\frac{a+b}{x+y} \right)$$
Now multiplying throughout by $-1$ (which flips the fractions in parentheses) ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Sum of the series $\tan^{-1}\frac{4}{4n^2+3}$ Find the value of $$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3}$$
I tried multiplying numerator and denominator by $n^2$, but got nothing. How do I split the term inside $\tan^{-1}$?
| I tried again and got the answer. $\frac{4}{4n^2 + 3} = \frac{1}{n^2 + 3/4} = \frac{1}{1 + n^2 -1/4} = \frac{1}{1 + (n+1/2)(n-1/2)} = = \frac{(n+1/2) - (n-1/2)}{1 + (n+1/2)(n-1/2)}$
$$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3} = \sum^{n=k}_{n=1}\tan^{-1}\frac{(n+1/2) - (n-1/2)}{1 + (n+1/2)(n-1/2)} = \boxed{\tan^{-1}(k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Finding the parameterization of a curve for a line integral problem I have to calculate the work of a particle that travel along a curve, given the following vector field:
$F(x, y, z) = (2z-1, 0, 2y)$
and where the curve is the intersection between:
$s1: z = x^2 + y^2$ and $s2: 4x^2 + 4y^2 + 1 = 4x + 4y$
using the defi... | There are many ways to parametrize a given curve, but I'll discuss one example here. The equation $4x^2 + 4y^2 + 1 = 4x + 4y$ can be rearranged:
\begin{align*}
4x^2 - 4x + 1 = -4y^2 + 4y -1 + 1
&\iff (2x-1)^2 = -(2y-1)^2 + 1 \\
&\iff (2x-1)^2 + (2y-1)^2 = 1.
\end{align*}
Knowing this, let us set $2x - 1 = \cos(t)$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives
Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that
$$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$
This isn't hard problem. I have already solved it in following way:
Let $x=\frac1a,y=\frac1b,z... | why must you use derivatives? the proof is simple with Cauchy Schwarz:
we have
$$\frac{1}{a^3(b+c)}=\frac{\frac{1}{a^2}}{a(b+c)}=\frac{\frac{1}{a^2}}{\frac{b+c}{bc}}$$ thus we have
$$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\geq $$
$$\frac{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}{\frac{2}{a}+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1292759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Verifying $\frac{\cos{2\theta}}{1 + \sin{2\theta}} = \frac{\cot{\theta} - 1}{\cot{\theta} + 1}$ My math teacher gave us an equality involving trigonometric functions and told us to "verify" them. I tried making the two sides equal something simple such as "1 = 1" but kept getting stuck. I would highly appreciate if som... | Use $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$ to get $$\frac{\cos(2x)}{1+\sin(2x)}=\frac{\cos^2(x)-\sin^2(x)}{1+2\sin(x)\cos(x)}$$
Multiply top and bottom by $\frac{1}{\sin^2(x)}$ to get $$\frac{\cot^2(x)-1}{\frac{1}{sin^2(x)}+2\cot(x)}$$
Using $\sin^2(x)+\cos^2(x)=1$ we can get $$\frac{\cot^2(x)-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1292982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative of $f(x) = \frac{\cos{(x^2 - 1)}}{2x}$ Find the derivative of the function $$f(x) = \frac{\cos{(x^2 - 1)}}{2x}$$
This is my step-by-step solution: $$f'(x) = \frac{-\sin{(x^2 - 1)}2x - 2\cos{(x^2 -1)}}{4x^2} = \frac{2x\sin{(1 - x^2)} - 2\cos{(1 - x^2)}}{4x^2} = \frac{x\sin{(1 - x^2)} - \cos{(1 - x^2)}}{2x^2} ... | You need the chain rule:
$$
\frac d {dx} \cos(x^2-1) = -\sin (x^2-1)\cdot\frac d{dx}(x^2-1)=\cdots.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1293272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Solving the Diophantine equation $x^n-y^n=1001$ For all $n \in \mathbb{N}$, solve the Diophantine equation $x^n-y^n=1001$, where $x,y \in \mathbb{N}$.
The cases $n=1,2$ are trivial ones. But for $n>2$ I can't find any solutions. How could I prove that there are no integer solutions for $n>2$?
| We'll$\let\leq\leqslant\let\geq\geqslant$ use the following, which can easily be proved using for example Bézout's theorem:
If $p$ is prime, then $p\mid x^n-y^n$ implies $p\mid x^{\gcd(n,p-1)}-y^{\gcd(n,p-1)}$.
Note $1001=7\cdot11\cdot13$ and $x^n-y^n=(y+(x-y))^n-y^n\geq(x-y)^n$. If $x-y$ and $n$ are large enough, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1294357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find the number of possible values of $a$
Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$,
and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$
Obviously, factoring,
$$(a-b)(a+b) + (c-d)(c+d) = 2010$$
$$a + b + c + d = 2010$$
Substituting you get:
$$(a-b... | Hint: examine the conditions you are given in the question - what can you say about the value of the expression in your last equation? Can you find a condition which makes it zero?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What mistake have I made when trying to evaluate the limit $\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b}$? Suppose $a$ and $b$ are positive constants.
$$\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b} = ?$$
What I did first:
I rearranged $\sqrt{n+a} \sqrt{n+b} = n \sqrt{1+ \frac{a}{n}} \sqrt{1+ \frac{b... | one way to see this is to use the binomial theorem. here is how it goes:
$$(n + a)^{1/2} = n^{1/2} + \frac12n^{-1/2}a + \cdots\\
\sqrt{n+a}\sqrt{n+b} = \left(\sqrt n + \frac a{2\sqrt n}+\cdots\right) \left(\sqrt n + \frac b{2\sqrt n}+\cdots\right) = n + \frac12(a+b)+\cdots $$
therefore $$\lim_{n \to \infty}\left(n - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 5
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How do I solve the trigonometric equation $1 - \sin^2x - \cos(2x) = \frac{1}{2}$? Solve for $x$ when $1-\sin^2x - \cos 2x = \dfrac{1}{2}$.
I can' t change it into a form I can work with. It is rather complicated.
| As given $$1-\sin^2 x-\cos 2x=\frac{1}{2} \implies \cos^2x-\left(2\cos^2 x-1\right)=\frac{1}{2}$$ $$ \cos^2 x=\frac{1}{2}=\cos^2 \frac{\pi}{4} \implies x=n\pi\pm \frac{\pi}{4}$$ Hence, the general solution of the equation is $x=n\pi\pm \frac{\pi}{4} $ where, $n$ is an integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1297353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$ I'm trying to evaluate this sum
$$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$
I have no idea how to deal with it.
With one sum I can, with partial-fraction decomposition, express it as a function of Digamma ... | As already pointed by the other answers, we just have to compute:
$$ S_1=\sum_{n\geq 1}\frac{1}{n^2+n-1},\qquad S_2=\sum_{n\geq 1}\frac{1}{(n^2+n-1)^2}.\tag{1}$$
and by exploiting the logarithmic derivative of the Weierstrass product for the cosine function we have, for any $\alpha\in\mathbb{R}^+\setminus\mathbb{N}$:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$
Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$
Is it correct if I consider the polynomial modulo $5$
$$(1+x)^{2015}=\sum\binom{2015}{n}x^n=1+2015x+2015\cdot1007x^2+\cdots+x^{2015}$$
RHS stays the same and then The remainder must be of the form $Ax+B... | use the division algorithm to write $$(1 + x)^{2015} = q(x)(x^2 + x+ 1)+ax + b\tag 1$$ the roots of $x^2 + x+ 1 = 0$ are $\omega = e^{i2\pi/3}, \bar {\omega}.$ we need $$1 + \omega = e^{i\pi/3},1 + \bar \omega = e^{-i\pi/3} $$ subbing $e^{i\pi/3}, e^{-i\pi/3}$ in $(1)$ gives us $$e^{\pm i2015\pi/3} = ae^{\pm i\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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General term of a sequence $(2-1)(2+1)(3-1)(3+1)...(n-1)(n+1)$ Can we use integrals, and are there some general methods for finding terms of a sequence?
| Let's reorder the terms and recognize the result:
$$
\begin{align}P&=(2-1)(2+1)(3-1)(3+1)\dots(n-1)(n+1)
\\&=(2-1)(3-1)\dots(n-1)\times(2+1)(3+1)\dots(n+1)
\\&=\frac{(n-1)!(n+1)!}{2}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$ Question:
Given $x + y = 12$, find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$?
Key:
I use $y = 12 - x$ and substitute into the equation, and derivative it.
which I got this
$$\frac{x}{\sqrt{x^2+4}} + \frac{x-12}{\sqrt{x^2-24x+153}} = f'(x).$$
However, aft... | You probably know about setting the derivative equal to $0$. The complexity of the equation we get may be discouraging. Your derivative is
$$\frac{x}{\sqrt{x^2+4}}-\frac{12-x}{\sqrt{x^2-24x+153}}.$$
I would rather write it as
$$\frac{x}{\sqrt{x^2+4}}-\frac{y}{\sqrt{y^2+9}}.$$
Nicer! Set this equal to $0$. So we get
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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$\int_{\Gamma} \frac{dz}{z}$, $\Gamma = \gamma([0,2\pi])$, $\gamma(t) = a\cos(t) + i b \sin(t)$ I am solving an exercise:
Evaluate by two methods: $$\int_{\Gamma} \frac{dz}{z}; \ \text{where:} \ \Gamma = \gamma([0,2\pi]), \text{and:}$$
$$\gamma: [0,2\pi] \longrightarrow \mathbb C \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ... | The first thing to do is use the double-angle formulae,
$$ \cos{2\theta} = 2\cos^2{\theta}-1 = 1-2\sin^2{\theta}, $$
which turns the denominator into
$$ \frac{1}{2} \left( (a^2+b^2) + (a^2-b^2)\cos{2\theta} \right) $$
Then we are dealing with something that has period $\pi$, so the transformation of the integral to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1310025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is this correct $(\cot x)(\sin x)=2(\cot x)^2$ $0≤x≤2\pi$ $x= \pi/2 , 3\pi/2$ $1.4371774 , 5.139467567$ Steps I took:
$$\cot x \sin x=2\cot^2 x$$
$$\cot x \sin x-2\cot^2 x=0$$
$$\cot x (\sin x-2\cot x)=0$$
$$\cot x \left(\frac{\sin^2 x}{\sin x} -2\frac{\cos x}{\sin x} \right)=0$$
$$\cot x \left(\frac{\sin^2 x-2\cos x}{... | Corrected solution:
Steps I took:
$$\cot x \sin x=2\cot^2 x$$
$$\cot x \sin x-2\cot^2 x=0$$
$$\cot x (\sin x-2\cot x)=0$$
$$\cot x \left(\frac{\sin^2 x}{\sin x} -2\frac{\cos x}{\sin x} \right)=0$$
$$\cot x \left(\frac{\sin^2 x-2\cos x}{\sin x} \right)=0$$
$$\cot x \left(\frac{1-\cos^2 x-2\cos x}{\sin x} \right)=0$$
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1310760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Roots of unity question. Question
Let $\omega=\cos\dfrac{4\pi}{7}+i\sin\dfrac{4\pi}{7}$. Show that
$\omega-1=2\sin\dfrac{2\pi}{7}\left(\cos\dfrac{11\pi}{14}+i\sin\dfrac{11\pi}{14}\right)$.
My attempt
Observe that $\omega$ is a seventh root of unity. Label the roots $1, \nu, \nu^2,\ldots,\nu^6$. Then $\omega=\nu^2$.
We... | I would do it differently. Recall identities
$$\cos 2x = 1 - 2\sin^2 x \\[1ex]
\sin 2x = 2 \sin x \cos x$$
to obtain
$$\cos \frac{4 \pi}{7} = 1 - 2 \sin^2 \frac{2 \pi}{7} \\[1ex]
\sin \frac{4 \pi}{7} = 2 \sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7}$$
and therefore
$$\omega - 1 = 2 \sin \frac{2 \pi}{7} \left( - \sin \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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What is the most unusual proof you know that $\sqrt{2}$ is irrational? What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive integer... | Ever so slightly off-topic, but I can't resist reminding folks of the proof that $\sqrt[n]{2}$ is irrational for $n \ge 3$ using Fermat's Last Theorem:
Suppose that $\sqrt[n]{2} = a/b$ for some positive integers $a$ and $b$. Then we have $2 = a^n / b^n$, or $b^n + b^n = a^n$. But Andrew Wiles has shown that there are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "114",
"answer_count": 19,
"answer_id": 10
} |
Is this a valid partial fraction decomposition?
Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions.
$$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$
$$4x+1 = A(x-2)+B(x+1)$$
$$x=2 \Rightarrow 4 \cdot2 + 1 = A(0) + B(3) \Rightarrow B = 3$$... | This is a classical source of confusion. But the the problem is avoided completely if one takes the limit of both sides as $x \to 2$, instead of substituting $x=2$.
Clearly the result is the same, but with this formulation there is no evaluation of functions at points where they are undefined.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1313454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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What is the spectrum of this matrix? $$A_n=\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 & 1\\
1 & 2 & 2 & \cdots & 2 & 2\\
1 & 2 & 3 & \cdots & 3 & 3\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & 2 & 3 & \cdots & n-1 & n-1\\
1 & 2 & 3 & \cdots & n-1 & n
\end{bmatrix}.$$
What are the eigenvalues and the corresp... | Since the inverse of $A_n$ is
$$
A^{-1}_n = \begin{pmatrix}
2 & -1 & 0 & \dots & 0 & 0\\
-1 & 2 & -1 & \dots & 0 & 0\\
0 & -1 & 2 & \dots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & 0 & \dots & 2 & -1\\
0 & 0 & 0 & \dots & -1 & 1
\end{pmatrix}
$$
Solving $A_n^{-1} x = \lambda^{-1} x$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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First contest problem I downloaded a contest and worked the first problem which is:
There exists a digit Y such that, for any digit X, the seven-digit number 1 2 3 X 5 Y 7 is not a multiple of 11.
Compute Y.
My attempt:
$$\begin{align}
1+3+5+7&=2+X+Y \\
16&=2+X+Y \\
14&=X+Y \\
\implies\quad Y&=14-X \\
\end{align}$$... | You say $Y =14-X$ if the number is divisible. Because X is a digit (in base ten), it ranges from $0$ to $9$; thus $Y$ ranges from $14-0=14$ to $14-9=5$. But note that $14$, $13$, $12$, $11$, and $10$ are not accepted values for a single digit. If the number has to be divisible by $11$, $Y$ must be $11$ less than the nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
solving difficult complex number proving if $z= x+iy$ where $y \neq 0$ and $1+z^2 \neq 0$, show that the number $w= z/(1+z^2)$ is real only if $|z|=1$
solution :
$$1+z^2 = 1+ x^2 - y^2 +2xyi$$
$$(1+ x^2 - y^2 +2xyi)(1+ x^2 - y^2 -2xyi)=(1+ x^2 - y^2)^2 - (2xyi)^2$$
real component
$$(1+ x^2 - y^2)x - yi(2xyi) = x + x^3... | If I started with your solution $$-2yx^2 +y + x^2 y - y^3 =0\implies y(x^2+y^2-1)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
How to find $\lim_{x \to 0}\frac{\cos(ax)-\cos(bx) \cos(cx)}{\sin(bx) \sin(cx)}$ How to find $$\lim_\limits{x \to 0}\frac{\cos (ax)-\cos (bx) \cos(cx)}{\sin(bx) \sin(cx)}$$
I tried using L Hospital's rule but its not working!Help please!
| i will expand on the hint given by dr. mv. we will compute the numerator and denominators separately. we have
$$\begin{align}\cos ax - \cos bx \cos cx &= 1 - a^2x^2/2 + \cdots - (1 - b^2x^2/2+\cdots)(1-c^2x^2/2+\cdots)\\&= \frac12 (b^2 + c^2 - a^2) x^2 + \cdots\\
\sin bx \sin cx &= (bx + \cdots)(cx + \cdots) = bcx^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
} |
Solving a linear congruence Use Euclids algorithm to find the multiplicative inverse of 11 modulo 59 and hence solve the linear congruence:
$11x \equiv 8 \mod59$
My working so far....
$ {11v + 51w = 1}$
Using Euclid's algorithm:
$ {59 = 5 \times 11 + 4}$
${11 = 2\times 4 + 3}$
${4 = 1 \times 3 + 1}$
$ {3 = 1 \times 3 ... | Good, so now you know $\displaystyle{59(3)+11(-16)=1}$. This tells you that
$$59(3)+11(-16)\equiv 1\pmod{\! 59}$$
$$\iff 11(-16)\equiv 1\pmod{\! 59}$$
Multiply both sides by $8$:
$$11(-16\cdot 8)\equiv 8\pmod{\! 59}$$
$$\iff 11(-128)\equiv 8\pmod{\! 59}$$
$$\iff 11(49)\equiv 8\pmod{\! 59}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1321970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent
First i subbed numbers in
$$\lim_{n \to \infty} \frac{(-1)^n}{1+\sqrt{n}} = \frac{-1}{1+\sqrt{1}} + \frac{1}{1+\sqrt{2}} - \frac{-1}{1+\... | Absolute convergence
If
$$ \sum \left|a_n\right| =\mbox{convergent}$$
Then
$$ \sum a_n =\mbox{absolutely convergent}$$
Note that an absolutely convergent series is also convergent.
Conditional convergence
If
$$ \sum \left|a_n\right| =\mbox{divergent}$$
And
$$ \sum a_n =\mbox{convergent}$$
Then
$$ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
System of equations with complex numbers-circles The system of equations
\begin{align*}
|z - 2 - 2i| &= \sqrt{23}, \\
|z - 8 - 5i| &= \sqrt{38}
\end{align*}
has two solutions $z_1$ and $z_2$ in complex numbers. Find $(z_1 + z_2)/2$.
So far I have gotten the two original equations to equations of circles,
$(a-2)^2 +(b-2... | The first thing to do is to make a drawing of the two circles. Label their centres $C_1$ and $C_2$, and the two intersections of the circles $I_1$ and $I_2$. Now connect these four points by straight lines, and label as $P$ the point half-way between $I_1$ and $I_2$, which is on the line connecting $C_1$ and $C_2$.
We... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"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.