1
vote
0answers
40 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
28 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
40 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 ...
0
votes
1answer
22 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
1
vote
1answer
41 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 ...
0
votes
0answers
39 views

integral of a sequence

Let $[a,b]$ be any closed interval in $\mathbb{R}$ such that $0\notin [a,b]$ and $f_n\in L^2(\mathbb{R})$ for $n\geq0$. If $\int_a^b f_n\phi\rightarrow \int_a^b f_0\phi$ for every $\phi$ in Schwartz ...
1
vote
1answer
21 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
57 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
47 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, ...
1
vote
1answer
43 views

Should I use the ratio test to determine convergence for $\sum_{k = 1}^{\infty}{1 \over k\left[1 + \ln^{2}\left(k\right)\right]}$?

I'm trying to determine whether this is convergent and I was wondering if using the ratio test would be the right way to do it? ${(k)(1+ln^2(k)) \over [k+1]\left[1 + \ln^{2}\left(k+1\right)\right]}$ ...
2
votes
1answer
62 views

Magical test for convergence of improper integrals?

I found this article while surfing the web. I hope it's not some kind of joke, because if it is it really fooled me. I'm trying to figure out the proof of theorem 2.3 I don't understand how the ...
9
votes
2answers
394 views

How prove this integral $\int_{0}^{\infty}f^{\alpha}(x)dx,\alpha>1$ is convergent

Question: let the function $f(x)\ge 0$,and such $$f'(x)\le\dfrac{1}{2},\forall x\ge 0$$ and this integral $\displaystyle\int_{0}^{\infty}f(x)dx$ is convergent. show that: ...
2
votes
1answer
47 views

Variant of dominated convergence theorem

There are several variants of dominated convergence theorem. The standard one requires $f_n \to f$ a.e. and $|f_n|\leq g$ a.e. where $g$ is integrable. It can be weakened to only convergent in ...
0
votes
2answers
28 views

Evaluate sequences of integrals with function bounded.

Evaluate $\lim\limits_{n\to\infty}n\int\limits_0^1 f(x)e^{-nx}dx$ where $\,f$ is bounded in $\mathbb{R}^+\cup\{0\}$. My problem is that I think there's information missing about $f$, e.g. some ...
0
votes
1answer
25 views

Convergence of Integrands and Integrals

Suppose $E \subset \mathbb{R}$ is compact. Is it possible to find a sequence of positive continuous functions $f_n: E \to \mathbb{R}$ such that for every $x \in E$ we have $$f_n(x) \to f(x)$$ for some ...
1
vote
1answer
51 views

Improper integral of Mixed Type Q

Q: Find the non-zero constant "c" such that the following integral is convergent. $$\int_{-1}^\infty \frac{e^{x/c}}{\sqrt{|x|}(x+2)}dx$$ Since the interval has both an infinite endpoint and ...
0
votes
1answer
44 views

Does a line integral depend continiously on the curve?

Let $\gamma_n: [0,1] \to \mathbb R^2$ be a family of curves which converges uniformly to the curve $\gamma$. Does the line integral $\int_{\gamma_n} \vec{F} \cdot \vec{d}s$ over an arbitrary vector ...
2
votes
1answer
60 views

Improper integral $\int^{1}_{0} \frac{x}{\sin{(x^{p})}} dx$

I have an improper integral with $p > 0$, $$\int^{1}_{0} \frac{x}{\sin{(x^{p})}} \ dx$$ and I want to find for which $p$ the integral exists. Now we should consider when $p = 1$ and when $p \not= ...
1
vote
1answer
31 views

Domination $\Rightarrow$ $0$ equality

Let $\phi \in C^{\infty}_c(\mathbb{R})$. In class, my teacher said that the dominated convergence theorem (DOM) may be used to prove that $$ \lim_{\epsilon \to 0^+} \int_{-\epsilon}^{\epsilon} \! \log ...
1
vote
0answers
29 views

Asymptotics of a convolution

For $r>1$ define the functions $$f(x)=|x|^{-1/2}\chi_{[-1,1]\setminus\{0\}}\quad\text{and}\quad g(x)=|x|^{-1/2r}(-\chi_{[-1,0)}+\chi_{(0,1]}).$$ I am interested in the asymptotic behavior of ...
1
vote
0answers
38 views

Range of values for which the integrals converge

I have two integrals (i) $\int_\gamma e^{z^2}dz$ where $\{\gamma: z=se^{i\alpha}: -\infty<s<\infty\}$ (ii) $\int_0^{\infty}\frac{x^\beta dx}{1+x}$ I know that the first integral converges for ...
1
vote
2answers
68 views

Finding a limit of an integral

I am trying to find the following limit. Let $X = [0,\infty)$ and $\mathbb B$ denote the Borel subsets in $[0,\infty)$, $\lambda$ the Lebesgue measure. Let $f_n : [0, \infty) \to \mathbb R$ be given ...
4
votes
2answers
100 views

Evaluate $I = ∫∫ 1/((x^2 + y^2)^{n/2}) dxdy$

Evaluate the double integral $$ I = \int\int_D \frac{1}{(x^2 + y^2)^{n/2}} dxdy .$$ where $n$ is an integer and $D$ is the region of the plane bounded by two circles centered on the origin and ...
1
vote
1answer
65 views

Twist on a well-known integral

It is well known that $\int_{\mathbb{R}^+} \sin (x^n)\,dx$ converges and has a closed form when $n>1$. Does $$\int_{\mathbb{R}^+}|\sin (x^n)|\,dx$$ converge for $n>1?$ My guess is that it does ...
3
votes
3answers
149 views

Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$?

$S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$ I stumbled on this question as an reading about Riemannian sums as in $$ \int_a^b f(x)\,dx =\lim_{x\to ...
10
votes
8answers
537 views

Proving convergence of a sequence whose terms are integrals

How to prove the following sequence converges to $0.5$ ? $$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$ What I have tried: I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n ...
0
votes
2answers
84 views

Does $\sum_{m=1}^{\infty}f_{m}\left(x\right) $ converge for almost every $x$ in $X$?

Let {$f_{m}$} be a sequence of measureable real-valued functions in $\left(X,\mathrm{\mathcal{M}},\mu\right)$. Suppose ${\displaystyle \sum_{m=1}^{\infty}\left\{ {\displaystyle ...
1
vote
0answers
14 views

Convergence of integrals in an arbitary Rieszspace

Hello i have a question about integration theory: The situation: Let $X$ be a set, $F$ a Rieszspace of $X$ and $\varphi$ an integral on $F$. Now we can extend this to the space of all integrable ...
0
votes
1answer
35 views

Do The Integrals Have a Majorant

Consider the integrals $\int_1^\infty\frac{x^2+kx} {x^4+k^px^2+k^2}dx$. For what p do the integrands have an integrable majorant? For what p do the integrals tend to $0$? I've seen other posts like ...
2
votes
1answer
87 views

Do The Integrals Tend to 0?

Consider the integrals $\int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x),$ where $m$ is the Lebesgue measure. For what $p$ do the integrands have an integrable majorant? For what $p$ do the integrals tend ...
3
votes
4answers
162 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
1
vote
2answers
60 views

Does this integral converge?

I am stuck on showing whether this integral converges or not: $$\int_0^\infty e^{\large{x/2-2\alpha(x^2-x^{1+\delta})}}\mathrm dx$$ where $\alpha>0$ and $0<\delta<1$. This seems pretty ...
2
votes
1answer
33 views

Convergence of functions in $L^2$ norm

Suppose $f\in L^2(\mathbb{R})$, and let $c>0$. Define $f_c(x)$ to be $f(x)$ when $|x|\leq c$ and $0$ when $|x|>c$. Show that $$\lim_{c\rightarrow\infty}\int_{-\infty}^\infty ...
1
vote
2answers
211 views

Convergence or divergence of integral

I'm struggling with how to show that $$ \int_1^\infty \frac{x \sin x}{\sqrt{1+x^5}}dx $$ either diverges or converges. If we call the integrand $f(x)$ then $$ f(x)\leq ...
1
vote
1answer
67 views

Why does this series diverge? $\sum_{n=1}^\infty \frac{n-1}{4n-1}$

So taking my original problem: $\sum_{n=1}^\infty \frac{n-1}{4n-1}$ I treated it like a limit problem as took the sum to be $\frac{1}{4}$ and since that is $<1$ for this geometric series, I ...
0
votes
1answer
42 views

What are some good integration problems where you can use some of the function convergence theorem of Lesbegue integrals?

I have learned about two major convergence theorem for the Lesbegue Measure: The monotone convergence theorem The dominated convergence theorem These are useful theorems for calculating integrals ...
2
votes
1answer
69 views

Confusion regarding proof using Fatou's lemma

This is in reference to the book Problems in Mathematical Analysis III by Kaczor and Nowak. We are given that ${f_n}$ converges to $f$ on $R$. Suppose that $\lim_{n \to \infty} \int_R{f_n dm} - \int_R ...
2
votes
2answers
68 views

Limit of integral of sequences

Calculate $$\lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx$$ I don't know how to calculate the integral and the sequence is not monotone or dominated by a $L^1$ function, ...
1
vote
1answer
112 views

Symmetric difference of sets and convergence in integration.

Let $(X,\mathcal{M},m)$ be a space of measure and $f_n,f \in L^1(m)$ such as $||f_n - f||_1 \rightarrow 0.$ Suppose that we also have $A_n,A \in \mathcal{M}$ and $m(A_n \triangle A) \rightarrow 0.$ ...
2
votes
1answer
496 views

Absolute and Conditional Convergence of the integral sin(x)/x^p for real values of p

I need to determine the values of p for which this integral converges conditionally and absolutely. $$\int_{0}^{\infty} \dfrac{\sin(x)}{x^p} dx $$ I think the interval for conditional convergence is ...
1
vote
1answer
43 views

showing that this integral is divergent

I am stuck on showing fornally that the following integral is divergent: for $n\in \mathbb{N}^+$, $\displaystyle\int_0^1 \frac{r^{n-1}}{(\lambda^2 + r^2)^{n/2}} dr $. Intuitively, if I take out ...
2
votes
1answer
90 views

Gamma Function Converges

I'm reading on pg 160 of Complex Analysis by Stein, and I am having trouble understanding the argument below -- it is intuitively plausible, but I don't see it rigorously. For s > 0, ...
4
votes
5answers
692 views

Convergence of integral $\int_{0}^{\infty}{\frac{1}{x^{3}-1}}dx$

So i have the integral $$\int_{0}^{\infty}{\frac{1}{x^{3}-1}} dx$$ Software programs say that it is divergent, except for one program which evaluate numerically and gave a clear result. The ...
5
votes
2answers
146 views

Convergence of $\int_0^1 \frac{\sqrt{x-x^2}\ln(1-x)}{\sin{\pi x^2}} \mathrm{d}x.$

I would like to prove the convergence of the Newton integral $$\int_0^1 f(x)\mathrm{d}x =\int_0^1 \frac{\sqrt{x-x^2}\ln(1-x)}{\sin{\pi x^2}} \mathrm{d}x.$$ I split this into two integrals ...
2
votes
4answers
83 views

Riemann Integrals converging to $0$

Let's assume that $\{f_n\}$ is a sequence of real,continuous,nonnegative functions on $[0,1]$ such that $$\int_0^1f_n(x)dx\overset{n\to\infty}{\longrightarrow} 0$$ I need to prove, or disprove that ...
0
votes
1answer
90 views

Absolute and conditional convergence of an integral

$\int_1^\infty x^p\cos(\ln x)\,dx$ Need to find values of $p\in \Bbb R$ for which the integral converges 1) absolutely and 2) conditionally. I've used partial integration and get that it converges ...
0
votes
0answers
69 views

A substitute for Dominated convergence (For swapping derivative and integral)

While reading a chapter on Z-Estimators from [1], I came across the following problem. (Note that the actual problem has nothing to do with estimation. I have posted the motivation at the end). Let ...
3
votes
2answers
132 views

Convergence of $\int_{-\infty}^{\infty} \sin p(x)\,dx$ where $p$ is a polynomial with $\deg p>1$

I thought about this problem today, and tried to solve it. Let $ax^n$ be the leading term of $p$. I can prove that $\displaystyle\int_{-\infty}^{\infty}\sin ax^n\,dx$ converges (below), and I argued ...
2
votes
4answers
86 views

Testing for convergence of this function

For the integral $$\int_2^\infty \dfrac{x+1}{(x-1)(x^2+x+1)}dx .$$ Can I know if it's convergent or not? If it does can I know how to evaluate it? I tried to use $u$ substitution but it didn't ...
5
votes
1answer
42 views

Integrability Question

A question that has popped up while studying for qualifying exams is the following: Prove that $\int_0^1 \int_0^1 \frac{1}{x^p + y^q} dx dy$ is integrable iff $p^{-1} + q^{-1} > 1$ I can handle a ...