2
votes
1answer
43 views

Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
0
votes
3answers
58 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
2
votes
3answers
90 views

Prove $\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$ using Dominated or Monotone Convergence

Is there a way to prove that $$\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$$ via the Dominated or Monotone Convergence Theorem?
3
votes
1answer
36 views

Prove the inequality based on an infinite series

Define $$f(x)=\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}.$$ It is easy to see that this series converges for $x\in(-1,1).$ Now we are asked to show that $(1-x)^2f(x)\geq x,$ for $x\in[0,1).$ I tried ...
4
votes
2answers
77 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
-2
votes
1answer
24 views

Sum of a convergent and divergent sequence [closed]

If $(x_n)$ is a convergent sequence and $(y_n)$ is a divergent sequence, show that their sum diverges.
-3
votes
0answers
37 views

Giving some hints [duplicate]

The sequence of $\begin{Bmatrix} {x}_{n} \end{Bmatrix}$ is strictly decreasing,$\quad \lim_{n\to\infty }{x}_{n}=0 \quad $,and$\quad \lim_{n\to\infty }{y}_{n}=0 .$ From the above ...
1
vote
2answers
41 views

uniform convergence of a functional sequence

Is this sequence of functions $$f_n(x)=n^3x(1-x)^n$$ converges uniformly for $x\in[0,1]$. I need some help on this.
8
votes
4answers
573 views

Can every positive real be written as the sum of a subsequence of dot dot dot

I answered this thing Infinite sum of prime reciprocals and now wonder what happens if we do not have such a strong condition as Bertrand's postulate. i have been fiddling with this, not sure either ...
0
votes
1answer
57 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
1
vote
1answer
93 views

A sum for stirling numbers Pi, e.

In this identity $$1-e{}^{2} = \displaystyle \sum _{n=0}^{\infty } \frac{(-1)^n(\pi )^{2 n}} {(2 n)!}\sum _{k=0}^{2 n} (-1)^{k} S_2(2 n,1-k+2 n),$$ $S_2$ is a Stirling number of the second kind. ...
1
vote
1answer
34 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
2
votes
1answer
59 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
-1
votes
4answers
145 views

Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge? [closed]

I need to solve $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$ Does this converge or diverge and why?
2
votes
1answer
72 views

How Find the sum $\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)$

Prove or disprove $$I=\sum_{n=2}^{\infty}\left(1+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots+\dfrac{1}{(2n-1)^2}\right) \binom{2n}{n}\dfrac{1}{2^{2n}(2n+1)}=\dfrac{\pi^3}{48}-\dfrac{1}{6}$$ My ...
5
votes
2answers
355 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
0
votes
0answers
26 views

geometric convergence of sequence

Let $\{x_n\}$ be a sequence of non-negative real numbers. For fixed $m \in \mathbb{N}$ Denote $$x_k^{(m)}:={1 \over m}\sum_{i=1}^m x_{k+1-i},\forall k\geq m$$ Assume $x^{(m)}_k\leq ...
0
votes
1answer
34 views

if $f(x)=\sum_{n=1}^{\infty}\frac{x^n}{n^2\ln{(n+1)}}$,show that $f$ is differentiable on $x=-1$ and $x=1?$..

let $$f(x)=\sum_{n=1}^{\infty}\dfrac{x^n}{n^2\ln{(n+1)}},-1\le x\le 1$$ prove $f$ is differentiable on $x=-1$ and $x=1?$.. for this problem I want show $f(x)$ is uniform convergence? maybe ...
0
votes
2answers
32 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...
0
votes
1answer
44 views

Analogue of differentiation for sequences?

I remember learning (2 semester calculus for engineers) about all the below ones, but nothing that fits in place of the question mark. Is there anything nontrivial? ...
0
votes
1answer
40 views

How do you formally prove $\limsup\limits_{n\rightarrow\infty}(|na_n|^{\frac{1}{n}}) = \limsup\limits_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}})$

My definition of $\limsup$ is it is the supremum of the accumulation points of a sequence (i.e the supremum of the limits of all possible subsequences of a sequence). So if: ...
2
votes
1answer
22 views

strengthen the condition of convergence in measure of sequence of functions

Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative measurable functions on a measurable set $E$. (1). Suppose for any $\epsilon>0$, $$\sum_{n=1}^\infty \mu\{x\in E: ...
1
vote
4answers
77 views

If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
1
vote
1answer
46 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
2
votes
2answers
59 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
81 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 ...
3
votes
0answers
69 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
3
votes
3answers
71 views

For $b \gt 2$ , verify that $\sum_{n=1}^{\infty}\frac{n!}{b(b+1)…(b+n-1)}=\frac{1}{b-2}$.

For $b \gt 2$ , verify that $$\sum_{n=1}^{\infty}\frac{n!}{b(b+1)...(b+n-1)}=\frac{1}{b-2}$$ This is how I tried.. ...
1
vote
2answers
65 views

Is the series uniform convergent in $(0,\infty)$?

For $$f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$ And is it bounded in $(0,\infty)$?
3
votes
2answers
67 views

What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
2
votes
3answers
60 views

Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
12
votes
2answers
180 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
1
vote
0answers
34 views

Abel's Functional Equation for $L(x) = \sum x^{n}/n^{2}$

In "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" Hardy talks of Abel's Functional Equation $$L(x) + L(y) + L(xy) + L\left(\frac{x(1 - y)}{1 - xy}\right) + L\left(\frac{y(1 - ...
0
votes
0answers
28 views

Proving a Cauchy Sequence without an explicit formula for $(a_n)$ [duplicate]

How would I prove that for all $n$ in the natural numbers, $|a_{n+1} - a_n| \le 1/2^n$ $(a_n)$ is a Cauchy Sequence? I have tried to use the standard definition and the triangle inequality, but ...
0
votes
1answer
28 views

A problem about convergence of sequences

Suppose $(a_n) $ is a bounded divergent sequence. I want to know if it is true that there is a point in the real line, and of course between the upper bound and the lower bound of the sequence such ...
1
vote
1answer
40 views

Prove that any unbounded sequence has a subsequence that diverges to $∞$.

To prove that any unbounded sequence has a subsequence that diverges to ∞, is it enough to say that you can take a subsequence $(a_{m(k)})$ where $m(k)=k$, as you know that this diverges to infinity, ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
9
votes
1answer
119 views

How prove $\sum_{n=1}^{\infty}\dfrac{a_{n}}{\ln{(1+n)}}<+\infty$

Question: Define sequence $\{a_{n}\}$ such $$0<a_{n}<1,n\in N^{+}$$ such $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{\ln{(a_{n})}}$$ is convergent. Show that ...
1
vote
2answers
31 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
1
vote
0answers
30 views

Proof regarding limits of functions and sequences

For a function f$:\mathbb{R} \rightarrow \mathbb{R}$ define what is meant by $ \ f(x) \rightarrow \infty$ as $ x \rightarrow -\infty$ and prove that it holds if and only if whenever ...
0
votes
1answer
38 views

What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
1
vote
2answers
81 views

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
0
votes
1answer
32 views

Prove that a sequence is decreasing

Suppose that 0 < a < 1 Show that {a^n} is a decreasing sequence. Yes, this is a homework question. I think I can solve it using induction, but I'm not sure.
1
vote
1answer
40 views

Small question about strong convergence

I have a small question I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
2
votes
1answer
65 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
1
vote
3answers
43 views

Question about sequences

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
0
votes
2answers
12 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
2
votes
0answers
66 views

Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

Does anybody know whether this limit here is zero, if we assume that $np_n \to \lambda$ for $n \to \infty$? $$ \mbox{So, do we have}\quad \lim_{n \to \infty}\sum_{k=0}^{n}{1 \over k!}\, ...
0
votes
2answers
36 views

Does this sequence converge to zero?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$, such that $\forall k \in \mathbb{N}: \lim_{n \rightarrow \infty} f(n,k) = 0$. Is it then true, that $\sum_{k=0}^n \frac{f(n,k)}{k!}$ converges to zero ...
1
vote
1answer
36 views

Why is this the possible Taylor series???

I am looking at an exercise at which it is asked to find the Taylor series of $f(x)=\log(1+x)$, $\xi=0, x \in (-1,1)$ $$f'(x)=(1+x)^{-1}$$ $$f''(x)=-1 \cdot (1+x)^{-2}$$ $$f'''(x)=2 \cdot ...