5
votes
3answers
121 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
0
votes
0answers
18 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
0
votes
1answer
27 views

Convergence uniformly implies in integral

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ such that ...
6
votes
0answers
75 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
1
vote
2answers
40 views

How to take the limit of the improper integral of a sequence of functions

Suppose $f_1, f_2, . . .$ are (Riemann) integrable functions. Then what is the $\epsilon$ definition of $$\lim_{n \rightarrow \infty} \lim_{M \rightarrow \infty} \int_{0}^{M} f_n(x) dx = L $$ for $L ...
9
votes
5answers
299 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
22
votes
1answer
388 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
9
votes
1answer
114 views

Interesting sum-integral equality

Is there an elementary proof of $$\lim_{n \to \infty} \int_0^\infty e^{-\alpha x^2} \frac{\sin((2n + 1)x)}{\sin x} dx = \pi\left(\frac{1}{2} + \sum_{k = 1}^\infty e^{-\alpha k^2 \pi^2}\right),$$ where ...
3
votes
1answer
104 views

Asymptotic behavior of a sequence of integrals

I am interested in the asymptotic behavior of sequences $(I_n)$ and $(J_n)$ as $n \rightarrow \infty$, where $$I_n = \int_{1}^{\infty}\frac{e^{-nx^2}}{x^2}\, dx,$$ and $$J_n = ...
2
votes
2answers
69 views

a question about summation of series, how to prove $\int_0^\infty e^{-x}S(x)$=$\sum_{i=0}^\infty a_nn!$

If the coefficients of $\sum_{n=0}^\infty a_nx^n$ is non-negative($a_n\ge 0$ for every n),and the sum function is S(x). Also,suppose$\sum_{i=0}^\infty a_nn!$ is convergent,please prove $\int_0^\infty ...
5
votes
5answers
89 views

a question about a complex integral, I am struggling with it!

How to prove $$\int _0^1 {\ln(x)\over{1-x^2}}={-\pi^{2}\over 8}$$ My solution: If we can prove$\int _0^1 {\ln(x)\over{1-x^2}}= \lim_{n\to \infty} \int _0^1\ln(x)(1+x^2+x^4+......+x^{2n})$,then I ...
0
votes
0answers
123 views

Taking the improper integral of a series term-by-term.

Suppose I have a function $f$ which appears in some complicated formula in a term that looks like $\int_{-\infty}^{\infty} f(x, v)\,\text{d}v$. Basically, I want to write $f$ in a series of functions ...
1
vote
1answer
73 views

Show that the improper integral $\int_1^\infty f(x) \ dx$ exists iff $\sum_1^\infty a_n$ converges.

The assignment: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and $f: [1, \infty) \rightarrow \mathbb{R}$ be a function, defined by $f(x) = a_n$, for $x \in [n,n+1).$ Show that: ...
0
votes
1answer
56 views

Convergence of the infinite series $2^x\ln(1+1/3^x)$

The Q: determine whether the series converges or not $$\sum_{k=1}^\infty 2^k\ln(1+1/3^k) $$ So far I figured out that the function is positive and decreasing on [1,infinity). I decided to try ...
0
votes
2answers
138 views

Integral test for convergence?

Why does the series' terms have to be non-negative to use the integral test? Consider the series: $$\sum_{n = 1}^{\infty}\frac{n\cos n - \sin n}{n^2}$$ Even though it has negative terms, why can't ...
2
votes
2answers
42 views

Is my divergence test correct?

This idea came to me while looking at the following graph of $f=\frac{1}{x}$: Now, the definite integral of $f$ from $1$ to $n$ is smaller than $f(1)+f(2)...f(n)$, from the graph above. But since ...
3
votes
2answers
81 views

Proving an inequality on $ \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$

EDIT : I have posted a proof below that needs to be reviewed. Some definitions Let $$\begin{array}{ccccc} f & : & \mathbb R_+^* & \to & \mathbb R_+^* \\ & & x & \mapsto ...
2
votes
1answer
99 views

Prove that $\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n!\zeta (n+1)$

I have encountered the following identity on Wolfram alpha and I fail proving it (with $n \in \mathbb N^*$) $$\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n! \zeta(n+1)$$ I tried to rewrite the ...
0
votes
2answers
78 views

Using integral test for $\sum 1/i^2$

It's been a while since I've done an Integral, but am required to relearn them for a class. Could anyone help me with the integral of $\dfrac{1}{i^2}$? Wouldn't the answer be $-1/i$ ? Context I'm ...
3
votes
1answer
77 views

Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
0
votes
2answers
85 views

Does the series from $1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge?

Does the series from $n = 1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge? Steps/tips would be greatly appreciated. Thanks!
0
votes
1answer
58 views

Equality between an improper integral and the sum of an infinite series

Let $a>0$ we want to show that $$\int_{0}^{\infty}\dfrac{\sin(t)}{e^{at}-1}=\sum_{0}^{\infty}\dfrac{1}{a^2n^2+1}$$ I assume that we want to find the power series expansion of ...
26
votes
2answers
696 views

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $f:[0,1]\to\mathbb R$ given by $$ f(x)=\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ ...
2
votes
0answers
53 views

Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
1
vote
1answer
49 views

Asymptotics of the logarithmic integral

Problem Given $$ \gamma = \int_0^1 {1-e^{-u} \over u} du - \int_1^\infty {e^{-u} \over u} du, $$ prove that $$ \int_0^x {dt \over \log t} = \gamma + \log \log x + \sum_{k=1}^\infty {\log^k x \over k ...
0
votes
1answer
145 views

Find an example of a positive function that its improper integral converge, but its series diverge

I need to find an example of a positive function f, and a constant a>0 such that the improper integral of f from 0 to infinity converge, but the series f(na) from 1 to infinity diverge.
3
votes
2answers
144 views

Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$

Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x = ...
0
votes
1answer
55 views

Is the series $\sum_{k=1}^{\infty}\frac{n_{k+1}-n_k}{n_{k+1}\log(n_{k+1})} $ convergent or not?

For an arbitrary monotone subsequence $\{n_k\}\subset \mathbb{N}^+$ and $\lim_{k\to \infty}n_k = +\infty$. Does $$\sum_{k=1}^{\infty}\frac{n_{k+1}-n_k}{n_{k+1}\log(n_{k+1})} $$ convergent or not? ...
13
votes
4answers
364 views

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums ...
0
votes
1answer
72 views

Is my reasoning correct about the convergence of this integral?

The integral is $\int_1^\infty\frac{\sin{x}}{x}dx$. I know that this integral converges, but I'm wondering if this is valid way to prove it. This function, if its domain is limited to ...
4
votes
1answer
86 views

Estimate a upper bound of an infinite series.

Assume $a>0$ and $a_n \geq 0$. how to verify that $$\sum_{n=1}^{\infty}\frac{a_n}{(a+S_n)^{3/2}}\leq \int_0^{\infty}\frac{1}{(a+x)^{3/2}}\mathrm{d}x$$ where $S_n = a_1+a_2+\cdots+a_n$ Thanks very ...
2
votes
4answers
583 views

Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$

$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$ I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
2
votes
2answers
57 views

What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$

What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$ I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
5
votes
1answer
115 views

Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
2
votes
2answers
53 views

Prove the following: Product of Roots

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ...
3
votes
0answers
140 views

Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Being inspired by this post, I've wondered if the infinite series below may be expressed as an intregral. I'm very curious about that. $$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$ ...
6
votes
3answers
325 views

A improper integral with Glaisher-Kinkelin constant

Show that : $$\int_0^\infty \frac{\text{e}^{-x}}{x^2} \left( \frac{1}{1-\text{e}^{-x}} - \frac{1}{x} - \frac{1}{2} \right)^2 \, \text{d}x = \frac{7}{36}-\ln A+\frac{\zeta \left( 3 \right)}{2\pi ^2}$$ ...
2
votes
1answer
92 views

Is there a closed form or a better solution?

This is another integral in the book "irresistible integral" I can find that: $$\int_0^\infty \frac{1}{\left( x^4 +2ax^2+1 \right)^{m+1}} \, \text{d}x =\frac{{{\left( -1 \right)}^m}\sqrt{2}\pi ...
7
votes
5answers
441 views

Definite integral, quotient of logarithm and polynomial

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
2
votes
1answer
310 views

An exponential improper integral

Evaluate : $$\int_{0}^{\infty }{\frac{{{\text{e}}^{-{{x}^{2}}}}}{{{\pi }^{2}}+{{\left( \gamma +x \right)}^{2}}}\text{d}x}$$
10
votes
2answers
225 views

A $\log$ integral with a parameter

Prove that: $$\int_0^\infty \frac{\ln x}{x^a+1}\;\text{d}x=-\left( \frac{\pi }{a} \right)\cot \left( \frac{\pi }{a} \right)\csc \left( \frac{\pi }{a} \right),\ \ a>1$$ For this one I consider to ...
36
votes
3answers
1k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\Large\frac\pi2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
10
votes
5answers
759 views

$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral

Anyone remember the method for this? I think this should been done on the site $$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$
-1
votes
1answer
85 views

Need help with these two integral

$$\displaystyle\begin{align} & \int_{0}^{\infty }{{{\text{e}}^{-x}}\left| \sin x \right|}\text{d}x \\ & \int_{0}^{\pi }{\cos \left( nx \right)\ln \left( 2\sin \frac{x}{2} \right)\text{d}x} ...
2
votes
0answers
185 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
2
votes
2answers
408 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
2
votes
2answers
117 views

Showing the divergence of $ \int_0^{\infty} \frac{1}{1+\sqrt{t}\sin(t)^2} dt$

How can I show the divergence of $$ \int_0^x \frac{1}{1+\sqrt{t}\sin(t)^2} dt$$ as $x\rightarrow\infty?$
4
votes
1answer
179 views

Infinite series representation. Limited or not?

Recently I've found the following. Let $n < m$. Then the integral $$\int\limits_0^\infty {\frac{{{x^{n-1}}}}{{1 + {x^m}}}} dx$$ converges and its value is (using the $B$ and $\Gamma$ function ...
10
votes
2answers
405 views

Convergence/Divergence of $\int_e^\infty \frac{\sin x}{x \ln x}\;dx$

I am currently doing some project and during the course of it I need to get an answer to the following: Does $\displaystyle \int_e^\infty \frac{\sin x}{x \ln x}\;dx$ converge/ absolutely ...
14
votes
3answers
386 views

Under what conditions is integrating over a series expansion valid for an improper integral?

On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...