6
votes
1answer
80 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 ...
3
votes
0answers
56 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
1
vote
3answers
57 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
1
vote
2answers
102 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
9
votes
5answers
240 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 ...
8
votes
0answers
89 views

A closed form for $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
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
4
votes
4answers
759 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
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 ...
4
votes
0answers
40 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
1answer
75 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 ...
2
votes
2answers
37 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
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 ...
2
votes
2answers
38 views

convergence of a sequence

I'm reading a paper, for proving a claim it defines $$ f_n(x) = \dfrac{(rx-x^2)^n}{n!} $$ when $ r = \frac{a}{b} $ is a rational, and $ I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx $ , and then it says ...
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
0answers
63 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
9
votes
1answer
104 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 ...
-1
votes
3answers
77 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
10
votes
2answers
122 views

Divergence of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
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
5
votes
2answers
133 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...
0
votes
2answers
34 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
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
5
votes
0answers
61 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
3
votes
2answers
68 views

$\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$

It is asked to prove: $\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$ I have tried to search for convergence and it gave me 0 so i can't solve it. ...
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: ...
2
votes
2answers
67 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
88 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
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 ...
0
votes
1answer
39 views

Integral Test and Series Convergence

Let $\alpha > 0 $. Then for what values of $\alpha$ does the following series converge? $$\sum_{k=1}^{\infty} \frac{1}{(k+1)[\ln(k+1)]^{\alpha}}$$ I believe the answer is the series converges if ...
0
votes
1answer
47 views

Compute the integral using Riemann sums.

Let $f(x)= 1 , \;-1\leqslant x\gt0,\; f(x)= 2,\; x=0,\; f(x)= 1 ,\; 0\lt x\leqslant1.$ Compute this integral using Riemann sums: $$\int_{-1}^1 f(x)\,\mathrm dx.$$ Any tips/solutions? I don't even ...
4
votes
5answers
116 views

What to do to calculate $\int_{-\pi/4}^{+\pi/4}e^{-\tan\theta}\mathrm{d}\theta$

I have to calculate the following integral: $$A=\int_{-\pi/4}^{+\pi/4}e^{-\tan\theta}\mathrm{d}\theta.$$ What I did: Let $t=\tan\theta$. Thus, ...
1
vote
2answers
137 views

Using integral test on $\sum_{n=2}^{\infty}\frac{1}{n^2 \ln n}$

As stated in the title, I have to use the integral test on $$\sum_{n=2}^{\infty}\frac{1}{n^2 \ln n}$$ to prove that it is convergent but I am having trouble doing that $$\lim_{b\to\infty} ...
1
vote
0answers
27 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
2
votes
0answers
22 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
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
1answer
26 views

Reciprocal Squareroot Birkhoff Integrable?

Is the reciprocal of the squareroot Birkhoff integrable over the unit interval: $$\int_{(0,1]}\frac{1}{\sqrt{x}}<\infty?$$ Then that would be an example of a function not Riemann but Birkhoff ...
0
votes
0answers
9 views

power series approximation to mertens function?

from the Mertens function $ \sum_{n=1}^{\infty}M(x/n)=H(x-1) $ if i take the Laplace transform to this equation i get $$ \frac{1-e^{-s}}{s}= \int_{0}^{\infty}dxe^{2xs}\frac{M(x)}{(e^{-xs}-1)^{2}}$$ ...
10
votes
1answer
236 views

Can I flip the integral and sum here?

I have the convergent integral-sum: $$\int_0^\infty \sum_{n\mathop=0}^\infty \frac {x^{4n+1}} {e^x - 1} \frac {(-1)^n} {(2n)!(4\pi)^{2n}}\mathrm d x$$ But is it the same as this?: ...
1
vote
0answers
66 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
1
vote
1answer
65 views

Derivatives of Series and the Fundamental Theorem of Calculus (Part 1)

The first part of the Fundamental Theorem of Calculus (FTC) states that: $$\frac{d}{dx}\int f(x)\,dx=f(x)$$ meaning that the indefinite integral of a function can be reversed by its equivalent ...
1
vote
1answer
69 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: ...
3
votes
1answer
50 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
1
vote
0answers
42 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
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
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 ...