Articles of calculus

How to solve $ \displaystyle\int \bigg(\small\sqrt{\normalsize x+\small\sqrt{\normalsize x+\sqrt{ x+\sqrt{x}}}}\;\normalsize\bigg)\;dx$?

How to solve this: $$\displaystyle\int \bigg(\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\sqrt{x}}}}\;\normalsize\bigg) \;dx$$

Finding values for integral $\iint_A \frac{dxdy}{|x|^p+|y|^q}$ converges

Given the following integral $$\iint_A \frac{dxdy}{|x|^p+|y|^q}$$ where $A=|x|+|y|>1$. How can one find for which $p$ and $q$ values the integral converges? Since the function is non-negative it is sufficient to show convergence/divergence on any Jordan exhaustion of $A$ in order to show convergence/divergence on $A$. I tried to use polar coordinates here, but I don’t […]

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}$

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}, b_{n+1} = \frac{a_n + b_n}{2}$. Show that $a_n, b_n$ converge to a common limit. Hint use inequality: $\sqrt{ab} \leq \frac{a+b}{2}$ attempt. Suppose $a_n\longrightarrow L_1$ and $b_n \longrightarrow L_2$ using the hint: $$\lim_{n \to \infty} 4a_nb_n < \lim_{n \to \infty} (a_n + […]

Understanding the solution to $\lim_{n\to\infty}\frac{a^n}{n!}, a>1$

Calculate $$\lim_{n\to\infty}\frac{a^n}{n!}, a>1$$ I need help understanding the solution to this problem: $$a_n:=\frac{a^n}{n!}$$ $$a_{n+1}=\frac{a^{n+1}}{(n+1)!}=\frac{a}{n+1}\cdot \frac{a^n}{n!}=\frac{a}{n+1}\cdot a_n$$ For $$n\geq n_0 :=\left \lfloor{a}\right \rfloor +1>a$$ we have $$\frac{a}{n+1}<\frac{a}{a+1}<1$$ $\Rightarrow$ the sequence $a_n$ is decreasing from $n_0$-th term onwards and obviously $a_n\geq 0, \forall n\in \Bbb N \Rightarrow a_n$ is convergent. Let $L:=\lim_{n\to\infty}a_n$. Then $$a_{n+1}=\frac{a}{n+1}\cdot a_n$$ $$L=\lim_{n\to\infty}\frac{a}{n+1}$$ $$L=0\cdot […]

Existence of a sequence with prescribed limit and satisfying a certain inequality II

This is a more specific variation of the question in the post Existence of a sequence with prescribed limit and satisfying a certain inequality Suppose you have two infinite sequences $\{a_n\},\{b_n\}$, with $0<a_n<b_n < 1$ for each $n$, such that both $a_n, b_n \to 1$ as $n \to \infty$. Further assume that $a_n/b_n \to 1$ […]

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method of doing this is for start from the outer variable ,in this case z, and from the original one can deduct that $z\in [0,1]$ But the professor goes on to make […]

Why $2n \int ^\infty _n \frac{c}{x^2 \log(x)} \sim n \frac{C}{n \log(n)}$?

I want to understand why we have $$ 2n \int ^\infty _n \frac{c}{x^2 \log(x)} \operatorname*{\sim}_{n\to\infty} n \frac{C}{n \log(n)} $$ where $c$ is a normalizing constant. I am unable to understand how the integral is removed.

$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$

For $a_n$ positive sequence. I think I can prove one direction, but not both.

evaluate the following integral

$$ \int J_0(x)\sin x~{\rm d}x $$ Where $J_0$ is Bessel function of first kind of order $0$ This what I tried $$ \int J_0(x)\sin x~{\rm d}x= -J_0(x) \cos x – \int J_0′(x)\cos x~{\rm d}x $$ $$ J_0′(x)=-J_1(x) $$ $$ \int J_0(x)\sin x ~{\rm}x= -J_0(x) \cos x -(J_1(x)\sin x – \int J_1′(x)\sin x~{\rm d}x) $$ $$ […]

PDE $(\partial_{tt}+a\partial_t-b\nabla^2)f(r,t)=0$

I am interested in solving the linear PDE for $f(r,t)$ $$ (\partial_{tt}+a\partial_t-b\nabla^2)f(r,t)=0 $$ $$ \nabla^2\equiv \frac{1}{r}\partial_r(r\partial_r)-\frac{1}{r^2}=\partial_{rr}+\frac{1}{r}\partial_r-\frac{1}{r^2} $$ with conditions $$ \frac{\partial f(0,t)}{\partial t}=0,\quad \frac{\partial f(R,t)}{\partial t}=d\cos (\omega t) $$ where $a,b,d,R>0$. You can see the laplacian like term is written in a cylindrical basis, so I assume Bessel function will arise. I would like a […]