14
votes
1answer
173 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
4
votes
0answers
56 views

Integral's Closed-form expression in terms of hypergeometric function

I want to solve the following integral: $$I = 2\left[\int_{0}^{1}\dfrac{y^m}{(1 - ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y+\int_{0}^{1}\dfrac{y^m}{(1 + ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y\right]$$ ...
4
votes
1answer
64 views

Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?

In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma $$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - ...
0
votes
2answers
55 views

A summation with binomial coefficients

Evaluate There seemed to be some problem with stackexchange's math rendering but Ian corrected whatever error was there in the expression.Thanks $$5050 \frac {\left( \sum _{r=0}^{100} \frac ...
1
vote
0answers
48 views

Differentiation and integration of a series

If I have a power series $$\sum _{k}^{\infty }f(x)$$ and I differentiate it I get according to my current knowledge $\sum _{k}^{\infty }f(x)'$,however when I look at a power series defined by $$\sum ...
1
vote
3answers
111 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
2
votes
1answer
62 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
20
votes
1answer
327 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 ...
1
vote
1answer
77 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
5
votes
3answers
160 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
3
votes
0answers
40 views

Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.

My problem is stated as follows Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define $ \displaystyle \hspace{3cm} I_n = \int_0^k ...
1
vote
1answer
79 views

Find limit $\lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right)$

I looked at the graph and found that limit is $\dfrac{1}{2}$ And limit to $-\infty$ is $-\dfrac{1}{2}$ By the way, the function for which we are finding the limit is called Fresnel function
2
votes
3answers
99 views

Prove that $s_n \leq 1+\ln n$, where $s_n$ is the $n$th partial sum of the harmonic series

This is a very Interesting question, there are many ways to do it. Lets see what is the best way to do it. I have an idea which involves a definite integral, I am working on it, will post it soon.
2
votes
4answers
118 views

Prove that $\lim\limits_{n \to \infty}\frac{x^n}{n!}=0$ [duplicate]

Well I can Intuitively see that. I am wondering If there is a neat way to prove that
18
votes
2answers
292 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
0
votes
0answers
35 views

Step in proof involving Euler-Mascheroni constant

I was just looking at a proof that shows how the Euler-Mascheroni constant exists and is situated between 0 and 1 . However, I stumbled across a step in the proof that doesn't seem very obvious to me ...
0
votes
1answer
54 views

Riemann integrable?

Consider the function: $f:[0,\frac{1}{2\pi}]\to\mathbb{R}:f(x):=x\cos(1/x)$ In general, every continuous function on a compact interval is Riemann integrable. However, for the tagged partitions: ...
0
votes
2answers
35 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): ...
5
votes
4answers
462 views

A sine integral

The integral \begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta \end{align} is claimed to not have a closed form expression. In this view find the series solution of the ...
2
votes
2answers
79 views

Proving that $\sum_{k=0}^\infty \frac{e^{ikb}-e^{ika}}{k}=i\int_a^b\frac{e^{it}}{1-e^{it}}dt$

I deleted my previous question because it was basically totally wrong. Let $a,b\in ]0,2\pi[$ Prove that $\displaystyle \sum_{k=0}^\infty ...
1
vote
1answer
25 views

Proving convergence/divergence for $p$-series

I have an exam in Calc 2 coming up. As such, I am doing previous exams given by our current professor. However, the exams lack a solution set, so I will post the question, and the answer I wrote down ...
11
votes
4answers
367 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
2
votes
2answers
120 views

Integral $ \int_{0}^{\infty} \frac{2axdx}{(x^{2}+a^{2})(e^{2\pi x}-1)} $

how could i evaluate the following integral ?= $$ \int_{0}^{\infty} \frac{2axdx}{(x^{2}+a^{2})(e^{2\pi x}-1)} $$ for positive 'a' ?? i have tried the expansion of the integran $ exp(2\pi x) -1 $ ...
0
votes
2answers
34 views

Calculus power series

Hi could anyone help me to solve this. express the function $\int_x ^0 (\sin(t^2)\cdot \cos(t^2))$ as a power series. Because there is two trigo identies I do not know how to combine them to form a ...
1
vote
0answers
38 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
0
votes
1answer
25 views

Integrating Trigonometric Series

Let $f(x)=c_0+c_1e^{i\theta}+c_2e^{2i\theta}+...+c_ne^{ni\theta}$ where $c_k\in \mathbb R$. We need to show $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
1
vote
1answer
57 views

Prove that : $\lvert s_n - \frac \pi 4\rvert \le \frac 1 {2n+1}$, where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1}$

Prove (Leibniz' series): $|s_n - \frac \pi 4| \le \frac 1 {2n+1}, \forall n \in \mathbb N$ where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1} = 1 - \frac 1 3 + \frac 1 5$ ... To prove the result ...
1
vote
1answer
72 views

Can $ \lim_{n \to \infty} \sum_{i=1}^{n} (1-\frac {x_i}{\sqrt{x_i^2+r^2}}) \cdot (x_{i+1}-x_{i})$ be written as a definite integral

$$ \lim_{n \to \infty} \sum_{i=1}^{n} \left (1-\frac {x_i}{\sqrt{x_i^2+r^2}}\right) \cdot (x_{i+1}-x_{i})$$ $x_1=1$, $x_n=a+L$ I don't really see a way to manipulate this into a desirable form, but ...
3
votes
2answers
58 views

How to find this limit using integration?

What is the value of $$\lim_{n \to \infty}\frac{(\sum_{k=1}^{n} k^2 )*(\sum_{k=1}^{n} k^3 )}{(\sum_{k=1}^{n} k^6)}$$ I just know that it has to be done by converting it into an integral. I have no ...
3
votes
3answers
99 views

What is $\displaystyle\lim_{n \to \infty} \space n^2\int_{0}^{1/n} x^{x+1} dx$?

How do we evaluate $$\lim_{n \to \infty} \space n^2\int_{0}^{1/n} x^{x+1} dx\quad ?$$ I know that $$\lim_{z \to 0+} \space \dfrac{\int_{0}^z x^{x+1} dx}{z^2}=\dfrac12,$$ and I think the asked ...
12
votes
4answers
241 views

Prove that $\displaystyle\int_0^1 x^a(1-x)^{-1}\ln x \,dx = -\sum_{n=1}^\infty \frac{1}{(n+a)^2}$

Prove that $$\int_{[0,1]} x^a(1-x)^{-1}\ln x \,dx = -\sum_{j=1}^\infty \frac{1}{(j+a)^2}$$ I know that we have a product of $x^a$, $\displaystyle\sum_{j=0}^\infty x^j$, and ...
0
votes
1answer
109 views

A new proof a non-linear Euler sum

According to Nielsen we have the following : If $$f(x)= \sum_{n\geq 0}a_n x^n $$ Then we have the following $$\tag{1}\int^1_0 f(xt)\, \mathrm{Li}_2(t)\, dx=\frac{\pi^2}{6x}\int^x_0 f(t)\, dt ...
24
votes
1answer
534 views

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

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, \\ ...
10
votes
8answers
547 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 ...
25
votes
2answers
488 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, ...
3
votes
1answer
140 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...
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} ...
2
votes
1answer
129 views

Evaluate limit of integral of sequence of function

Evaluate $$\lim\limits_{n\to \infty}\int_{0}^{1}\frac{\sqrt{n}(e^{-x/n}-1)}{x}dx.$$
1
vote
3answers
321 views

Evaluating $\lim_{n\rightarrow\infty}x_{n+1}-x_n$

Let $f(x)$ be continuously differentiable on $[0,1]$ and $$x_n = f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+f\left(\frac{3}{n}\right)+\ldots+f\left(\frac{n-1}{n}\right)$$ Find ...
24
votes
2answers
1k views

Integrating $\int_0^ex^{1/x}\;dx$

Compute $$\int_0^ex^{1/x}\;\mathrm dx.$$ There is an analytical anti-derivative found in this answer. How does one compute this? Using the anti-derivative approach we have $$\int x^{1/x}\;\mathrm d ...
3
votes
2answers
145 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 ...
0
votes
1answer
183 views

Integral of a series [closed]

I cannot solve this integral, can anyone help me? $$\int_0^\infty \left(x^3 \sum_{n=1}^{+ \infty} e^{-nx} \right)dx$$ Thank you in advance
13
votes
4answers
338 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 ...
2
votes
0answers
67 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
3
votes
2answers
114 views

Show that $\int_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}$

I'm looking for a clever way to show that $$ \int\limits_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}.$$ All suggestions will be appreciated!
5
votes
1answer
248 views

Qualifying problem for real analysis: limit involving definite integral

The following problem has appeared in 2013 January qualifying exam in Purdue University, which is publicly available here. Problem 3. Let $\{a_k\}$ be sequence of positive numbers such that ...
4
votes
0answers
71 views

Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,du$

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,du.$$ The parameters are possibly complex, and satisfy $$\Re(c)>-1, ...
5
votes
1answer
118 views

Limit of an integral related to the beta function: $\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}$.

Consider the following limit: $$ Z(\beta) = \lim_{z\to1-}\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}. $$ (This is related to this question.) What is the closed form for this limit? ...
9
votes
1answer
261 views

Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y}$?

Consider the following integral: $$G(\beta,\delta,y) = \int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y},$$ with $\delta>0$, $\Re\beta>0$, $y\neq1$. Does it have a closed form in ...
14
votes
2answers
316 views

A Challenging Euler Sum $\sum\limits_{n=1}^\infty \frac{H_n}{\tbinom{2n}{n}}$

Recently, I encountered a strange series involving Harmonic Numbers and Binomial Coefficients both. According to Mathematica: $$\displaystyle \sum_{n=1}^\infty \frac{H_n}{\binom{2n}{n}} = ...