Articles of calculus

Looking for examples of Discrete / Continuous complementary approaches

Among many fascinating sides of mathematics, there is one that I praise, especially in teaching at undergraduate level: the parallels that can be drawn between a “Continuous world” and a “Discrete world”. A concept, an explanation, etc. in one world that can be “reflected” into the other world. More an help to a global understanding… […]

Calculate the integral $\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt$, by deformation theorem.

I want to prove: $$\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt=\frac{2\pi}{ab}$$ by the deformation theorem of complex variable. Then I consider a parameterization $\gamma:[0,2\pi]\rightarrow A$, traveled in the opposite direction of the clock hand, of the ellipse (in $\mathbb{C}$): $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ I thought of searching for a function $f:A\rightarrow \mathbb{C}$ (analytic in $A$) and a curve $\lambda:[0,2\pi]\rightarrow A$ (homotopic to $\gamma$) and […]

Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?

I’ve had no luck with this one. None of the convergence tests pop into mind. I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I know that $\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n} \to 0$ but not sure if it’s decreasing. Regarding absolute convergence, I tried: $$|\sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}|\geq \sin^2 n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}=$$ $$=\frac{1}{2}\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}-\frac{1}{2}\cos 2n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$$ But again […]

Question about arithmetic–geometric mean

We have two sequences: $$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$ I need to prove that those are making Cantor’s Lemma.(At the end I shold get that: $\lim_{n\to \infty}a_n=\lim_{n\to \infty}b_n$ by Cantor’s Lemma) Any ideas how? Thank you.

“Why do I always get 1 when I keep hitting the square root button on my calculator?”

I asked myself this question when I was a young boy playing around with the calculator. Today, I think I know the answer, but I’m not sure whether I’d be able to explain it to a child or layman playing around with a calculator. Hence I’m interested in answers suitable for a person that, say, […]

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by this question, I sought to find $($a justification for$)$ the closed form expressions of the following two integrals: $~\displaystyle\int_0^\infty\frac{J_A(x)}{x^N}~dx~$ and $~\displaystyle\int_0^\infty\frac{J_A(x)~J_B(x)}{x^N}~dx.~$ For the former, we have $~\displaystyle\int_0^\infty\frac{J_{2k+1}(2x)}{x^{2n}}~dx~=~\frac12\cdot\frac{(k-n)!}{(k+n)!}~,~$ for $k>n>\dfrac14~,~$ which I was ultimately able to “justify” $($sort of$)$ in a highly unorthodox manner, using […]

Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$

Find the limit without the use of L’Hôpital’s rule or Taylor series $$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$

A limit problem related to $\log \sec x$

If $$f(x) = \dfrac{{\displaystyle 3\int_{0}^{x}(1 + \sec t)\log\sec t\,dt}}{(\log\sec x)\{x + \log(\sec x + \tan x)\}}$$ then prove that $$\lim_{x \to {\pi/2}^{-}}f(x) = \frac{3}{2}$$ and $$\lim_{x \to 0}\frac{f(x) – 1}{x^{4}} = \frac{1}{420}$$ Looking at the integral sign in numerator I see that the best way to attack this problem is via L’Hospital Rule. But that […]

How do I determine $\lim_{x\to\infty} \left$?

How do I determine this limit ?: $$ \lim_{x \to \infty}\left[x – x^{2}\log\left(1 + {1 \over x}\right)\right] $$ I have tried to decompose the $\log$ function but I don’t know how to proceed from there.

If $f$ is continuous and $f(x+y)=f(x)f(y)$, then $\lim\limits_{x \rightarrow 0} \frac{f(x)-f(0)}{x}$ exists

I’m solving the functional equation $f(x+y)=f(x)f(y)$ and I know that I have a continuous function $f:[0,\infty\rangle \to \langle 0,\infty\rangle$ s.t. $f(0)=1$. In one of the steps, I want to show that the limit $$\lim_{x \rightarrow 0} \frac{f(x)-1}{x}$$ exists and is finite? I’m just looking for a hint.