1
vote
0answers
17 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
58 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 ...
8
votes
1answer
96 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
74 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
9
votes
2answers
105 views

Divergent 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
78 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
126 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
29 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
93 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
112 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
55 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
67 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
50 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
60 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
83 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
34 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
460 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
35 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 ...
1
vote
1answer
46 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
135 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
25 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
354 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
119 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
234 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
61 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
64 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
66 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
41 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
33 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 ...
11
votes
1answer
298 views

$-4\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)=S$

EDIT: Due to the solution below, I edited the answer of the post. Thanks!!!! Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty ...
2
votes
1answer
51 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
2
votes
1answer
80 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
1
vote
1answer
44 views

Prove $\ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{10n}\leq \ln\big(\frac{20}{9}\big).$

Please help. I have been trying his question for the past 2 hours and cant seem to go anywhere with it. $$ \ln\big(\frac{21}{10}\big) \leq \sum_{n=10}^{20} \frac{1}{n}\leq \ln\big(\frac{20}{9}\big). ...
2
votes
3answers
74 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 ...
4
votes
0answers
78 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
1
vote
2answers
48 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? ${(k)(1+ln^2(k)) \over [k+1]\left[1 + \ln^{2}\left(k+1\right)\right]}$ ...
0
votes
1answer
53 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 ...
2
votes
3answers
57 views

standard Taylor series using substitution

Find Taylor series using substitution about $0$ for $f(x)=\frac{125}{(5+4x)^3}$ by writing $\frac{125}{(5+4x)^3}=\frac{1}{(1+\frac{4}{5}x)^3}$? Determine a range of validity for this series.
2
votes
1answer
75 views

Generalized sophomore's dream

It is well know that $$\int_0^1 x^{-x} dx=\sum_{n=1}^{\infty}n^{-n}.$$ I'm wondering if there is a way to characterize all the continous functions $f: \mathbb{R}^{+}\rightarrow \mathbb{R}$ such that ...
3
votes
3answers
123 views

Series Expansion Of An Integral.

I want to find the first 6 terms for the series expansion of this integral: $$\int x^x~dx$$ My idea was to let: $$x^x=e^{x\ln x}$$ From that we have: $$\int e^{x\ln x}~dx$$ The series expansion of ...
0
votes
2answers
112 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 ...
1
vote
0answers
35 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 ...
1
vote
1answer
55 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 ...