11 votes
Accepted

Is it valid to use L'Hopital's rule inside a power?

This isn't about l'Hôpital's rule, it's a much more basic property of limits. Let $g(x)$ be a function such that its limit as $x\to x_0$ equals $p\in\mathbb R$. That is, let $\lim_{x \to x_0} g(x) = ...
NtLake's user avatar
  • 339
8 votes

$F(x)+F(\frac{1} {x})=\frac{3}{2}$ where $F$ is an antiderivative of $f$

Let's look at the form $p(x)=F(x)+F\left(\frac{1}{x}\right)$. We have $$\begin{align}p'(x)&=f(x)-\frac{1}{x^2}f\left(\frac{1}{x}\right)\\ &=\frac{\log x}{x+1}-\frac{1}{x^2}\cdot \frac{\log \...
Chris Lewis's user avatar
  • 2,061
8 votes

After integrating $\csc^2(x)\cot(x)$ why do I get $\cot^{2}(x) = \csc^{2}(x)$?

They differ by a constant: \begin{align*} \cot^{2}(x) = \frac{\cos^{2}(x)}{\sin^{2}(x)} = \frac{\cos^{2}(x) + \sin^{2}(x)}{\sin^{2}(x)} - 1 = \frac{1}{\sin^{2}(x)} - 1 = \csc^{2}(x) - 1 \end{align*}
Átila Correia's user avatar
6 votes
Accepted

Acrobatic Calculus of "Physicists" in deriving the Taylor series for $e^x$

Here's a more rigorous version - not completely, but it highlights what rigour requires. Let $J_af$ denote $\int_a^xf(t)dt$ so $(I-J_a)e^x=e^a$, where $I$ is the identity (I could have written $1-J_a$,...
J.G.'s user avatar
  • 116k
6 votes
Accepted

How to approximate $1.05^{50}$ by hand

The answer is not so close to $1$. $1.05^{20}$ is approximately $e=\lim\limits_{n\to\infty}\left(1+\dfrac1n\right)^n$. $1.05^{50}$ is approximately $e^{2.5}$, which is $\sqrt{e^2e^3}$. $e\approx2.7$, ...
J. W. Tanner's user avatar
  • 60.6k
6 votes

If $\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right)$

Put $L_n = \sum_{k=1}^n \frac{k}{n} \log\Big(\frac{n^2 + (k-1)^2}{n^2 + k^2}\Big)$, for the $n$th term, and define $f(x) = \log(1+x^2)$. Then, \begin{align*} L_n &= \sum_{k=1}^n \frac{k}{n} \log\...
Drew Brady's user avatar
  • 3,690
6 votes

Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

For $f\neq 0$ the left hand side is equal $\infty,$ as $$\int\limits_{-\infty}^\infty|f(x)|^2\left [\int\limits_{-\infty}^\infty{dy\over |x-y|^\alpha}\right ]\,dx\\ =\int\limits_{-\infty}^\infty|f(x)|^...
Ryszard Szwarc's user avatar
5 votes

How to solve $\int_0^\frac{\pi}{4} \frac{x \cos^2(2x)}{(1+\sin(2x)) (1+ \cos(2x))} dx$?

I'll just do the last two integrals you got to: For the first you can apply IBP choosing $u=x$ and $\mathrm dv=\dfrac{\mathrm dx}{1+\cos x}$: $$\begin{aligned} I_1\equiv\int_0^{\pi/2}\!\!\!\!\frac{x}{...
Joan S. Guillamet F.'s user avatar
5 votes

Is there a closed form expression for $ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $?

Not an answer: $$\sum _{n=2}^{\infty } \frac{1}{n \sqrt{n^2-1}}=\\\int_0^{\infty } \left(\mathcal{L}_n^{-1}\left[\frac{1}{n \sqrt{n^2-1}}\right](x)\right) \sum _{s=2}^{\infty } \exp (-s x) \, dx=\\\...
Mariusz Iwaniuk's user avatar
4 votes

Acrobatic Calculus of "Physicists" in deriving the Taylor series for $e^x$

Interpret $f\to \int f = \partial^{-1} f$. By the fact, the the constant functions are in the kernel of $\partial,$ $\partial c =0$, $\int $ is the pseudo inverse, that, at each application adds a ...
Roland F's user avatar
  • 2,172
4 votes

If $\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right)$

We can note that if $f(x) =\log(1+x^2)$ then $f'(x) =2x/(1+x^2)$ and hence by mean value theorem $$\log\frac{1+(k-1)^2/n^2}{1+k^2/n^2}=-\frac{1}{n}\cdot\frac{2t_k}{1+t_k^2}$$ for some $t_k\in((k-1)/n,...
Paramanand Singh's user avatar
  • 87.4k
4 votes

If $\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right)$

Let $f(x) = -\log(1+x^2)$. Then the sum inside the limit can be recast as: $$ \sum_{k=1}^{n} \frac{k}{n}\left[ f\left(\frac{k}{n}\right) - f\left(\frac{k-1}{n}\right) \right] \to \int_{0}^{1} x \, \...
Sangchul Lee's user avatar
4 votes
Accepted

Question from Born, Optics, p. 188 about system of two thin lenses

Taking the variation $\delta f$ of a function satisfies the chain rule: \begin{equation} \delta (f\circ g) = \delta g \frac{df}{dg} \end{equation} Use this, linearity, and the Leibniz rule to see (...
Toby Saunders-A'Court's user avatar
3 votes

Determining if this interval is true or false. $\int_{-2}^2\left(x^5-9x^{11} + \frac{\sin(x)}{(1+x^4)^2}\right)dx = 0$?

The function you want to integrate is odd which means that its integral over a symmetric interval is equal to 0.
Kosmas Nikopoulos's user avatar
3 votes

Finding $\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$

Substitute $t=\tan(\frac\pi4-\frac x2)$ \begin{align} I=&\int_0^{\frac{\pi}{2}}\tan^{-1}\left(\sin x\right)dx\\ =&\int_0^1 \frac2{1+t^2}\tan^{-1}\frac{1-t^2}{1+t^2}\ dt \overset{ibp}=4\int_0^1 ...
Quanto's user avatar
  • 97.5k
3 votes
Accepted

Showing that $ f(x) = 0 $ if the triple integral $\frac{f(x)}{(1+x^2+y^2+z^2)} $ converges

I'm not sure that will work since the range of the $a$ and $b$ variables will shrink as you go outwards, so that the integrals in those variables might go to zero. It might be better to use that ${\...
Zarrax's user avatar
  • 45k
3 votes

How to approximate $1.05^{50}$ by hand

$\lim_\limits{n\to \infty} (1+\frac {1}{n})^n = e$ $(1+\frac {1}{n})^n < e$ for finite $n$ so this will give us an upper bound. Let $n = 20$ $(1+\frac {1}{20})^{20} \approx e$ $1.05^{50} \approx e^{...
user317176's user avatar
  • 11.2k
3 votes
Accepted

$\int \sin e^x dx$: Where did I go wrong?

After you obtained $$\int \sin e^x \, dx = \int \frac{1}{z} \sin z \, dz,$$ it is unclear how you applied integration by parts. Presumably, you chose $$u = \frac{1}{z}, \quad dv = \sin z \, dz.$$ ...
heropup's user avatar
  • 136k
3 votes
Accepted

A generalisation of $\lim\limits_{x\to0}\frac{e-(x+1)^{\frac{1}{x}} }{x}$.

Ommiting the $e$, numerators and denominators form respectively sequences $A055505$ and $A055535$ in $OEIS$. If you look at the Mathematica section, you will find ...
Claude Leibovici's user avatar
3 votes
Accepted

Proof check: prove that if f is derivable in $x_0$ then $\ \lim_{x\to x_0} f(x) = f(x_0)$

If $f$ is derivable in $x_0$ then it exists $\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0} = L \in \mathbb{R}$. So $$\lim_{x \to x_0} (x-x_0) \cdot \lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}=0$$ which ...
MathLearner's user avatar
3 votes

Solve for $x$: $x^2 + x = 14\log_2(\log_2(x)) + 6, \quad x > 1$

You have $$x^2+x-6=(x+3)(x-2)=14\log_2(\log_2(x))$$ Note that you have $\log_2(\log_2(x))=0$ when $x=2$ and $\log_2(\log_2(x))=1$ when $x=4$ (i.e. when $(x+3)(x-2)=14$). These two facts give the two ...
Piquito's user avatar
  • 29.7k
3 votes
Accepted

Is there a closed form expression for $ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $?

Here are a few more integral representations. Firstly, one can use the complex integral representation for $\binom nk$ and the generating function of $\zeta(2n)$ from the question: $$\sum_{n=2}^\infty ...
Тyma Gaidash's user avatar
3 votes

Deal with discontinuity in double integrals

$f$ is obviously not well-defined at $(0,0)$, and we can show that $$\lim_{(x,y) \to (0,0)} f(x,y)$$ does not exist for any path in $D$, since if $y = kx$ for some $k \in [0,1]$, we have for $x > 0$...
heropup's user avatar
  • 136k
2 votes

On the solutions of $y''=yP(x)$ or $y'+y^2=P(x)$

I looked up the four coefficient sequences of the sums in the OEIS. The first sum is elementary and the other three can be expressed as ${}_2F_1$'s: $$\sum_{n=0}^\infty\binom{3n}nx^n=\sum_{n=0}^\infty\...
Parcly Taxel's user avatar
2 votes

Integral of $\int_{-r}^r e^{\beta y} \arctan\left(\frac{\sqrt{r^2-y^2}}{d}\right) dy$

Here's a partial answer: First, integrate by parts to yield $$\frac{d}\beta \int_{-r}^r \frac{y e^{\beta y} \,dy}{(d^2 + r^2 - y^2) \sqrt{r^2 - y^2}} .$$ The substitution $y = r \sin \theta$ gives $$\...
Travis Willse's user avatar
2 votes

"Sandwich" Quadratic majorizer of even convex function

A counterexample: $$ f(x) = \max(|x| - 2, 0) \, ,\\ p(x) = (x-1)^2 \, . $$ $f$ is convex, even, with $f(0) = 0$ and $f(x) \le p(x)$ for all $x \in \Bbb R$. If $q$ is an even quadratic polynomial ...
Martin R's user avatar
  • 113k
2 votes
Accepted

A formula for $\pi$

I can think of a few things a mathematician might care about including: beauty, computational complexity, convergence, and sign of the error. Beauty: This is highly subjective, though it oftentimes ...
Eric's user avatar
  • 6,443
2 votes

Comment on points of maxima and minima for $ f(x) = \frac{\sin(\pi x)}{x^2} $

The derivative is $$f'(x)=\frac{\pi x \cos (\pi x)-2 \sin (\pi x)}{x^3}$$ Because of the leading $x$, the solution will be closer and closer to $ \left(n+\frac{1}{2}\right)$. Let $x=\left(n+\frac{1}...
Claude Leibovici's user avatar
2 votes

How to evaluate $\int_0^1 \frac{x^n}{x^2+x+1} \,dx$

Note that $x^2+x+1=(x-e^{i\frac{2\pi}3})(x-e^{-i\frac{2\pi}3} )$ and $$\int_0^1 \frac{x^n}{x-a}dx =a^n\ln\frac{a-1}a+\sum_{k=0}^{n-1}\frac{a^k}{n-k} $$ Then, with $a= e^{i\frac{2\pi}3}$ \begin{align} &...
Quanto's user avatar
  • 97.5k
2 votes

definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$

Note that \begin{align} I=&\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx\\ =&\int_{0}^{\frac{\pi}{4}} \frac{(\sin x\cos x)^2(\sin x+\cos x)}{(\sin x+\cos x)^2(\sin^2x-\sin ...
Quanto's user avatar
  • 97.5k

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