Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

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5
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4answers
70 views

Proving that the series $\sum_{k=2}^\infty \frac{1}{k \ln k}$ diverges?

I don't know how to show this. The terms go to zero, and I can't really show that the terms dominate $\frac{1}{k}$ (a series with these terms diverges). Any other ideas?
1
vote
3answers
48 views

Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge

This is a homework problem for an undergrad topology course. Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in ...
0
votes
0answers
18 views

Expand function using Maclaurin's series(infinite form)

Expand the function f(x)=log(1+x) in powers of x in an infinite series stating the validity of such expansion for x belonging to (-1,1]. The question actually asks to show that cauchy's remainder or ...
0
votes
3answers
57 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
0
votes
0answers
51 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...
3
votes
1answer
53 views

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? Is my solution correct?

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? I am confused because my friend insists the series converges conditionally. I think the series diverges. Here is my process and solution: ...
0
votes
1answer
31 views

Difficulty understanding Divergence Test

I'm studying Series and Diverge Test. But I'm having a problem understanding it. It says that, when the limit of it's partial sums is not equal to zero then it diverges. But then, there's also an ...
0
votes
3answers
37 views

Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge?

The integral is $\int { (1+n^2)^{-1/4}}dn$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$\sum (1+n^2)^{-1/4} \cos ...
0
votes
4answers
63 views

Does the series $\sum \frac{1}{n\ (\ln(n))^{3/2}}$ converge or diverge?

Consider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but ...
28
votes
2answers
1k views

Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent?

As I was just checking this 'child prodigy' out on Youtube, I stumbled upon this video, in which Glenn Beck asks the kid to do the following proof: Further on, the kid starts sketching a proof ...
0
votes
2answers
91 views

Does the equality $1+2+3+… = -\frac{1}{12}$ lead to a contradiction? [duplicate]

Is $1+2+3+4+5.... = -\frac{1}{12}$ self-contradictory ? I've heared much that $1+2+3+.... = -\frac{1}{12}$, although the fact that this series is diverging. I saw a proof of it by a physicist. In ...
1
vote
1answer
69 views

Divergence of $u_{n+1}=1+\frac{n}{u_n}$

Let $u_n$ be defined by $u_0=1$ and $u_{n+1}=1+\frac{n}{u_n}$. It can be shown easily that if it has a limit, then it must be $+\infty$. Does $u_n$ diverge to $+\infty$ ? What I have tried : Let ...
2
votes
0answers
28 views

How to find the analytic continuation of this series?

I have the following series: $$ \sum_{n = 0}^{+\infty} \frac{n^2}{(n^2 + a^2)^{\epsilon}} $$ with $a\in \mathbb{R}$. How can I find its analytic continuation for $\epsilon \in \mathbb{C}$? In ...
1
vote
1answer
41 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
1
vote
1answer
22 views

How to test convergence for a tetration series slightly below the harmonic series?

I have the following series to test convergence \begin{align} S_{\infty}= \sum_{n=1}^{\infty} \dfrac{1}{n} \left( \dfrac{1}{n} \right)^{ \left( \dfrac{1}{n} \right) } < \sum_{n=1}^{\infty} ...
0
votes
1answer
83 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
0
votes
0answers
31 views

Why does Cauchy's Root Test for convergence of infinite series require $\limsup$?

I'm confused about the reasoning behind Cauchy's root test for convergence of infinite series. It states that for any series $\{a_n\}$, if $C = \limsup_{n\rightarrow\infty}{\sqrt[n]{|a_n|}} < 1$, ...
1
vote
1answer
22 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
2
votes
1answer
30 views

Convergence study of a series of functions

I am studying the convergence of the series $$ \sum_{n=0}^{\infty}\frac{\sin (x^n)}{(1+x)^n} $$ where $x \in \mathbb R$. My initial approach was to use the ratio test, but I am not getting to ...
4
votes
1answer
71 views

Is it possible to sum the divergent series with prime coefficients?

This is a follow-up of this question. It is known that the divergent series $$ P := \sum_{n=1}^\infty p_n \qquad \text{where } p_n \text{ is the $n$th prime} $$ cannot be summed by means of (prime) ...
0
votes
2answers
40 views

Determining if series converges or diverges

The Series is For this series the ratio test is inconclusive. I have rewritten the series as Currently i am approaching the problem using limit test. I couldn't progress from this point. Any ...
0
votes
0answers
6 views

Upper and lower Abel sums for bounded sequences

Let $a_k,k\geq 0$, be a bounded sequence and consider the ``upper and lower Abel sums": $UA(a):=\limsup_{x \to 1-}\sum_{k=0}a_kx^k$ and $LA(a)=\liminf_{x \to 1-}\sum_{k=0}a_kx^k$. Is it true that ...
4
votes
1answer
81 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
8
votes
1answer
86 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
0
votes
3answers
71 views

Convergence of the series $\sum n!/(n^2+3)$

How can we test if this series diverges/converges? $$\sum_{n=1}^\infty\frac{n!}{n^2+3}$$ I tried D'Alembert's principle and tried to do $\frac{a_{n+1}}{a_n}$ but I'm stuck. Any help?
3
votes
0answers
166 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
0
votes
1answer
78 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ [closed]

I need to find out whether this series converges or diverges. $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ Can someone help how to solve it?
1
vote
2answers
32 views

Convergence test of certain series

I need to find out whether this sequence converges or diverges using limit comparison test. $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$ I've tried it with the use of sequence ...
3
votes
2answers
96 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
0
votes
0answers
53 views

Strange sum divergent

Could you Find /Check the strange sum i have calculate in a unformal way it is like a analitycal continuation $$\sum _{k=1}^{\infty } (-1)^k k \log \left(\frac{k+3}{k+2}\right)=\frac{1}{6} (-36 \log ...
1
vote
3answers
90 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
3
votes
5answers
141 views

Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
3
votes
3answers
110 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
6
votes
0answers
153 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
0
votes
0answers
27 views

Find the closed-form of a series

Suppose that $x$ is positive number such that $x>0$. I just wonder is there existing a closed form of the series below $f(x)=\sum_{l=0}^{\infty}(2l+1)e^{-xl(l+1)}$. Is the well-known ...
3
votes
2answers
44 views

Simple series divergence problem

I've got a problem here: $$\sum_{n=1}^{\infty} \frac{5^n}{n(3^{n+1})}$$ I've used the ratio test and essentially did this: $$\sum_{n=1}^{\infty} \left( \frac{5^{n + 1}}{n (3^{n+1+1})} / ...
-1
votes
2answers
71 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [closed]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
5
votes
2answers
289 views

Is there a group-theoretic proof of the Riemann rearrangement theorem?

The analytic proofs of the Riemann rearrangement theorem are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I ...
2
votes
2answers
91 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
1
vote
2answers
89 views

How to tell if a log series converges?

I have the following series. $$(-1)^n \times \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)$$ I tried the root, ratio and integral tests, but am doing something wrong because I am unable to tell if this series ...
1
vote
1answer
105 views

Playing fast and loose with divergent series [closed]

I have been playing around recently with the regularization of infinite divergent sums and products, e.g. $$1+1+1+1+1+\ldots=\zeta(0)=-\frac{1}{2}$$ $$1+2+3+4+5+\ldots=\zeta(-1)=-\frac{1}{12}$$ ...
2
votes
2answers
43 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
5
votes
4answers
117 views

Is $\sum\limits_{n=1}^\infty \sin{\frac{(-1)^{n+1}}{n}}$ convergent?

$$ \mbox{Is}\quad \sum_{n=1}^\infty \sin\left(\left[-1\right]^{n + 1} \over n\right) \quad\mbox{ convergent ?.} $$ $$ \mbox{I know that }\quad\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\quad \mbox{is ...
1
vote
1answer
31 views

Trying to understand the math in a neuroscience article by Karl Friston

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
0
votes
1answer
28 views

$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
1
vote
1answer
27 views

Two cases involving Maclaurin Series

Could you help me to prove it? I'm working hard in it, but I got nothing.
4
votes
1answer
71 views

References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign ...
1
vote
0answers
50 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
0
votes
1answer
63 views

Concerning the sum $\sum_{n = 1}^\infty \sin nx$

I recently came across this question and I posted an answer. It has been pointed out that my answer is incorrect. I cannot work out what is wrong with my reasoning. The answer I gave corresponds with ...
1
vote
2answers
57 views

Divergence of modified harmonic series

I am reading a paper which claims that the following series diverges: $\sum\limits_{n=2}^{\infty}\frac{1}{nH_{n-1}}$ where $H_{n}$ is the $n$'th harmonic number $\sum\limits_{m=1}^{n}\frac{1}{m}$. I ...