3
votes
1answer
44 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
15
votes
2answers
177 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
0
votes
2answers
95 views

Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
0
votes
1answer
19 views

Determine whether this integral converges for 0 < p < 1?

$$ \int_{0}^{a} \frac{1}{x^p} $$ I'm lost on how to approach this problem. So far I've integrated it: $$ \int_{0}^{a} \frac{1}{x^p} = \lim_{b\to0}\frac{x^{1-p}}{1-p} \int_{a}^{b} $$ But I don't ...
0
votes
1answer
18 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
0
votes
3answers
50 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
0
votes
1answer
37 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
1
vote
4answers
102 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
1
vote
0answers
25 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
10
votes
4answers
166 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
1
vote
2answers
32 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
2
votes
3answers
131 views

Decide convergence divergence of $\sum \dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Does the series $\sum \dfrac{1}{(\ln n)^{\ln n}}$ converges? I can intuitively say that it converges, because $(\ln n)^{\ln n} $ is going to $\infty$ on a hayabusa
4
votes
0answers
44 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
1
vote
0answers
38 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
0
votes
2answers
45 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
2
votes
5answers
178 views

Is integral convergent?

I have a problem with following integral: $\int_1^\infty \frac{\sqrt{x}}{1+x} \sin(2x)dx$ I was trying to prove convergence (or divergence) of this integral, however without any success. My best ...
0
votes
1answer
29 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
0
votes
2answers
61 views

How we compare the two following integral without calculation?

compare the two following integral without calculation : 1)$\displaystyle{\int_0^1x{e^{x^2}}dx}$ 2)$\displaystyle{\int_0^1 \sqrt{x}{e^{x}}dx}$ I would be interest for any comments or any replies
11
votes
1answer
172 views

The integral on $[0,1]\times[0,1]$

Here I have a problem. $p$ and $q$ are positive numbers. the integral $$\int_0^1\int_0^1 \frac{1}{x^p+y^q}\;dx\;dy< \infty \Longleftrightarrow \frac{1}{p}+\frac{1}{q}>1$$ Here is my try, ...
1
vote
1answer
53 views

Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?

If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = ...
0
votes
1answer
26 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
1
vote
1answer
35 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
2
votes
2answers
92 views

convergence of a generalized Riemann integral

Could you please provide me some hints to test the convergence of this integral below? $$\int\limits_{0}^{+\infty}\dfrac{\sin x\cdot \sin 2x}{x^\alpha} \, dx$$ where $\alpha \in \mathbb{R}$
1
vote
2answers
75 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
0
votes
1answer
59 views

Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...
1
vote
1answer
68 views

Check the uniform convergence of parametric integral

While $0< \alpha <+ \infty$, prove if the parametric integral is uniform convergent on $\alpha$'s domain: $$\int^{+ \infty}_{0} e^{- \alpha x} \sin \beta x dx$$ $\beta$ is nonzero constant.
1
vote
2answers
37 views

I want to prove a series converges absolutely

I want to show that: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}}$ converges absolutely. Now, here is what I have done: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}} \leq ...
0
votes
2answers
45 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
2
votes
1answer
49 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
0
votes
1answer
20 views

Proving that the harmonic p series converges for p>1 and diverges for p<=1

Can someone please check if I have done this correctly? The harmonic p-series: $$ \sum_{n=1}^\infty \frac{1}{n^p}$$ $$ let $$ $$f(n)=\frac{1}{n^p}$$ $$ f(x)=\frac{1}{x^p}$$ Since f(x) is a ...
3
votes
2answers
94 views

How to calculate the improper integral $\int_0^\infty\left(\frac{1}{\sqrt{x^2+4}}-\frac{P}{x+2}\right)dx$

This is the first time I've seen a problem like this. I have no idea what to do. Detailed guidance would be of great help. For which values of P does the integral converge? ...
1
vote
3answers
53 views

Comparison test for $\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$

$$\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$$ How come up with something to compare it to use the comparison test?
2
votes
1answer
37 views

Integral comparison test

$$ \int_a^b \frac{x+1}{\sqrt[4]{x^5+x^2}} \,dx $$ I want to know this integral converges or not when $(a, b) = (0, 1), (1, \infty)$. I was thinking of using the comparison test, but I can't think of ...
1
vote
0answers
26 views

Solve a problem of convergence of integral

We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$. How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial ...
2
votes
1answer
93 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
1
vote
1answer
24 views

Indefinite integral convergence

How can I prove that this integral converges? $$ \int_0^1 \sqrt{\frac{1-kx^2}{1-x^2}} dx\quad\quad 0\leq k<1 $$ Edit: fixed typo dt -> dx
2
votes
1answer
60 views

Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
2
votes
0answers
55 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
1
vote
2answers
61 views

Check if $\int_1^{\infty\:}\left(e^{-\sqrt{x}}\right)dx$ converges using Convergence Test

I could use some help with an homework question: Using the convergence test, check if the following integral function converges or diverges (no need to calculate the limit itself): ...
1
vote
1answer
30 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
1
vote
1answer
77 views

Show that $\lim_{x\to\infty} \frac{1}{x} \int_0^x f(x) \ dx = \gamma$.

The Assignment Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a function which is integrable on the intervall $[0,x] \ \forall x > 0$ and $\lim_{x\to\infty} f(x) = \gamma \in \mathbb{R}$. ...
3
votes
1answer
54 views

When $\int_{0}^{\infty}f(x)dx=\sum_{n=0}^{\infty}\int_{n}^{n+1}f(x)dx$?

Is the following always true? (i.e. if both converges, limits are equal; if one diverges, the other must diverge; EXCLUDE the case where the limit keeps "jumping") $$ ...
2
votes
0answers
37 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
1
vote
0answers
61 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
0
votes
1answer
32 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
0
votes
1answer
50 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
1
vote
1answer
48 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
1
vote
1answer
23 views

what a and b make the integral convergence?

Consider the following integral $$\iint_Ax^\alpha y^\beta \space dA$$ where $A=\{(x,y)\space|\space0\leq y\leq1-x,x\geq0\}$. Find all possible values of $\alpha$ and $\beta$, for which this integral ...
2
votes
3answers
90 views

Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that ...
1
vote
2answers
50 views

Not sure which test to use?

Trying to determine if the following series is convergent: $$\sum_{k = 1}^{\infty} {2^k ln(1+1/(3^k))}$$ I have no idea how to compute the integral so im not sure if I should use the integral test, ...