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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4
votes
2answers
39 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
67 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
271 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
83 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
85 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
95 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
0answers
101 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
2
votes
2answers
38 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
107 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
30 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
0answers
59 views

$1+2+3+4+5… = -1/12$ [duplicate]

$1+2+3+4+... = -1/12$ I know it has been asked a million times but can someone reply an "understable" answer for non mathematician ? which one is true ? both true, both false ? $1+2+3+4+... = -1/12$ ...
0
votes
1answer
26 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
55 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
49 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
60 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
50 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 ...
0
votes
1answer
41 views

Prove or disprove the convergence of…

I need help with the following problem, please help. For positive real x. Let $${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$ Prove or disprove the convergence ...
0
votes
1answer
73 views

Convergence of $\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$

$\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$ A) For which $p\in \mathbb{R}$ is the series convergent? B) For which $p\in \mathbb{R}$ is the series divergent, and what is ...
1
vote
2answers
30 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
0
votes
0answers
23 views

Define $u_n$ and $v_n$ inductively

it is asked to find an expression for $u_{n}$ in terms of n and for $v_{n}$ also in terms of n $a, b \in $R $v_1=(a+b)/2$ $u_1=\sqrt{ab}$ $u_{n+1}= \sqrt{u_n\cdot v_n}$ ...
3
votes
1answer
25 views

Discuss the convergence or the divergence of the series

$x_n=($ln $n)^{-p}$ is the nth term of the series I am working on. I have tried looking at the series for different ranges of $p$. I also have noted that the ratio test is inconclusive. My work: If ...
8
votes
2answers
188 views

What is the sum of all complex integers?

In line with $$\zeta(-1)=-1/12$$ Could we, by considering $$f(s)=\sum_{a,b\in\mathbb Z,\;(a,b)\neq(0,0)}\frac{1}{(a+bi)^{s}}$$ Evaluate the sum of all complex integers?
8
votes
0answers
105 views

To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees. The idea to construct such a bijection stems from looking at some ...
19
votes
1answer
251 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
0
votes
2answers
41 views

Divergent test for a simple series

how would I be able to prove that, using the comparison test, diverges? Using symbolab gave me diverges, but it does not show how, and it used the series root test, which I will not cover in my ...
1
vote
0answers
36 views

Euler product for Riemann zeta and analytic continuation

the Euler product for the Riemann zeta $$ \zeta (s)= \prod _{p}\left( \sum_{n=0}^{\infty}p^{-ns}\right) $$ this is only valid for $ \Re(s) >1 $ however we could use the Borel transform so $$ ...
0
votes
1answer
31 views

Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
38
votes
1answer
976 views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
0
votes
1answer
46 views

Divergence Test Question

How would I show this series diverges $$\sum_{r=1}^{\infty} \frac{(-1)^rr^3}{2r^3+3r^2+1}$$ It's a monotonically increasing sequence, so i know the series would diverge, but how would i prove this? ...
3
votes
2answers
44 views

Proof divergence $\sum_{n=1}^\infty (\frac{n+1}{n})^n$

Proof divergence $\sum_{n=1}^\infty (\frac{n+1}{n})^n$ I don't know how to do this problem, maybe a hint or two will help.
-4
votes
1answer
32 views

Question on convergent and divergent sequences [closed]

Is every divergent sequence constant? Please provide an example. Thanks!
0
votes
1answer
21 views

Evaluate the following infinite series or state that the series diverges.

From my textbook. $$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$ My work: So a constant greater than or equal to $1$ raised to ∞ is ∞. A number $n$ for $0<n<1$ is $0$. So when taking the ...
1
vote
1answer
62 views

Convergence or Divergence of a Series Using Case Analysis

In the problems below it's asked for which $r \in \Bbb R$ the series converges. $$ a)\quad\sum_{k=0}^\infty \left( \left(\sum_{l=1}^k \frac1l\right) r^k\right) $$ $$ b)\quad\sum_{k=0}^\infty ...
1
vote
0answers
24 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
0
votes
2answers
71 views

Prove this series is divergent: $\sum_{k=1}^{\infty}\sin kx$

I need to prove that $\sum_{k=1}^{\infty}\sin kx$ is divergent when $x \notin \pi \Bbb Z$. I tried to solve this equation with it's sums' seria but I didn't succeed. I'll be glad for some help.
0
votes
4answers
50 views

how to find out series is divergent or convergent for $\sum_{n=1}^\infty \frac{2^n}{n^2}$

\begin{equation}\sum_{n=1}^\infty \frac{2^n}{n^2}\end{equation} The text book says the above series diverges by the n-th term test, but given no procedures how it was done so, could you some ...
4
votes
3answers
174 views

Can the sum $1+2+3+\cdots$ be something else than $-1/12$?

I've seen methods to calculate this sum - also in questions on this site. But it seems it is a matter of how you want to regularize the problem. Are there summation methods which could give a ...
1
vote
3answers
46 views

$\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent..

I am stuck on the following problem that says: Let $\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent. Then it follows that a. $a_n \to 0$ but ...
0
votes
1answer
15 views

What convergence test can I use on this series?

I am doing practice problems for an exam, and I am not sure how to test this series: Limit from n=1 to infinity of cos(n) * sin^2(1/n) I am supposed to use lim x -> 0 sin(x)/x = 1 to find the ...
0
votes
2answers
31 views

Find any sequence that meets these criteria.

I'm struggling with this problem and don't know where to start looking: Is there any sequence $a_n$ such that $\lim\limits_{n \to \infty}a_n \neq 0$ and $\lim\limits_{n \to \infty}(n \sqrt[n]{|a_n|}) ...
0
votes
2answers
24 views

Power Series — Convergence, Divergence, and Absolute Convergence

Suppose that the power series $$\sum a_nx^n$$ is convergent at $x=-3$ and divergent at $x=5$. What can be said about the following: convergence at $x=-2$ ? absolute convergence at $x=2$ ? ...
0
votes
1answer
20 views

Linearity of summation of divergent series

I just learnt that one of the axioms of a summation method for a divergent series is linearity: $$S[\sum_{n=0}^{\infty}(\alpha a_{n} + \beta b_{n})] = \alpha S[\sum_{n=0}^{\infty} a_{n}] + \beta ...
1
vote
2answers
77 views

Prove that $\ln(x)$ diverges

Prove that $\ln(x)$ diverges using the fact that the harmonic series diverges. How can I compare the $\ln$ with the harmonic series, if the harmonic series appears to be more relevant to the ...
1
vote
1answer
58 views

Value of divergent series?

Let $\{a_n\}$ be a positive, convergent sequence. We consider the sequence of partial sum $\{s_n\}: s_n = \sum_{k=1}^n a_n$. Clearly $\{s_n\}$ is strictly increasing and therefore $\sum_{n=1}^\infty ...
1
vote
1answer
20 views

Limit Comparison Question

I have a interesting problem in my book. It states: Show that if $a_n > 0$ and $\lim\limits_{n\to\infty} (n \cdot a_n) \neq 0$, then $\sum a_n$ is divergent. It hints at using limit comparison ...
1
vote
1answer
44 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
0
votes
2answers
33 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
1
vote
4answers
100 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
3
votes
2answers
28 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...