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Questions tagged [divergent-series]

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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How to prove $\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz$ in the sense of Borel summation?

As the title shows, I would like to prove this identity in the sense of Borel summation, $$\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz,$$ providing ...
HC Zhang's user avatar
2 votes
2 answers
176 views

Does the rate $\log n$ imply "almost harmonic"?

Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$ I have found a reference where it says that the converse is true. I tried to tackle ...
L--'s user avatar
  • 814
1 vote
2 answers
78 views

Alternating series comparison test

Let's say I have two alternating series of terms, $(-1)^n A_n$ $(-1)^n B_n$ If I know (by for example Leibniz criteria) that one of the series converges / diverges, can I use comparison criteria to ...
Simeon Stefanović's user avatar
1 vote
1 answer
64 views

$\sum 2^{-r_n}/r_n$ diverges $\implies$ $\sum 2^{-\lceil r_n \rceil} / {\lceil r_n \rceil}$ diverges

I want to prove $\sum 2^{-r_n}/r_n$ diverges $\implies$ $\sum 2^{-\lceil r_n \rceil} / {\lceil r_n \rceil}$ diverges where $r_n$ is a nondecreasing sequence of reals. This came up in Billingsley ...
tail_recursion's user avatar
1 vote
1 answer
47 views

Elliptic integral singular expansion

The question. Consider the Elliptic Integral $$ F(x;k)=\int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}.\tag{1}\label{1} $$ I am interested in the singular series expansion of $F(1;k)$ about $k=1$. I was ...
Gateau au fromage's user avatar
2 votes
3 answers
103 views

Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$

I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$. In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
Mariod's user avatar
  • 71
2 votes
1 answer
69 views

Differently defined Cesàro summability implies Abel summability

I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
Kangyeon Moon's user avatar
0 votes
0 answers
70 views

the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $

I was looking at the zero's of $$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$ for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$. Now this clearly relates to the Riemann zeta: $$f(s,a) + \...
mick's user avatar
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2 votes
0 answers
80 views

Does the limit of the exponential mobius exponential series asymptotically equal its regularized power series?

Context: Consider the function $\sum_{n=0}^{\infty} e^{nx}$. An extremely unrigorous manipulation of this series would yield $$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
Sidharth Ghoshal's user avatar
0 votes
0 answers
13 views

Show that $\sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}$ with $ \epsilon > 0$ diverges [duplicate]

I think this sum diverges but I can't seem to show it. $$ \sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}, \ \ \ \epsilon > 0 $$ I have tried using the bound $n \log n = \log n^n < n^n$ which led me ...
Ryderr's user avatar
  • 133
0 votes
1 answer
42 views

"Boundary" between convergent and divergent series of the form 1/n^m.

Since $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, but $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, is there some $m \in \mathbb{R}$, with $1 < m < 2$ that defines the "boundary" ...
spacecowboy's user avatar
0 votes
1 answer
49 views

Cesaro $(C,\alpha)$ summable implies Abel summable.

I've found quite a few questions regarding the statement "Cesaro $(C,1)$ summability implies Abel summability", e.g. this question, but haven't been able to find a proof for higher Cesaro ...
Jonathan Huang's user avatar
0 votes
3 answers
60 views

Show that for $(u_{n})$ strictly decreasing sequence tending to 0, $\sum_{n=0}^{+\infty} \frac{u_{n}-u_{n+1}}{u_{n+1}}$ diverges

I come to ask you about a problem coming from a serie's exercice sheet brought by a student that I can't crack. Let $(u_{n})$ be a strictly positive decreasing sequence which converges to 0. How do I ...
Armand Jourdain's user avatar
3 votes
0 answers
41 views

How can we discuss the "divergent-ness" of an infinite series?

There are many infinite series that converge "regularly" to a finite value, such as the geometric series $$ \sum^\infty_{n=0}\frac{1}{2^n} = 1+\frac{1}{2} + \frac{1}{4} + \cdots = 2. $$ ...
Jonathan Huang's user avatar
1 vote
0 answers
13 views

How to prove Exercise 8.2.6 from Analysis 1 Terence Tao [duplicate]

I have been stuck a while on the following exercise of Analysis 1 from Terence Tao: Exercise 8.2.6 Let $\sum^\infty_{n=0}a_n$ be a series which is conditionally convergent, but not absolutely ...
Smogogole's user avatar
0 votes
0 answers
95 views

How to find the divergent renormalization of $\sum_{n=1}^{\infty} \frac{2^n}{n}$?

Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n} $$ This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n} $ and 'evaluating' that power ...
Sidharth Ghoshal's user avatar
4 votes
2 answers
70 views

Showing divergence of $\sum\limits_{k=1}^{\infty} \log\left(1+\frac{(-1)^{k+1}}{k^\alpha}\right)$ where $0<\alpha<\frac{1}{2}$

I am trying to prove that $$\sum\limits_{k=1}^{\infty} \log\left(1+\frac{(-1)^{k+1}}{k^\alpha}\right)$$ diverges, for any $0<\alpha<\frac{1}{2}$. I tried showing this by taking the Taylor ...
Ofek Levy's user avatar
0 votes
2 answers
111 views

Are Caesaro summation and Abelian summation frown upon in mathematical community?

If we agree to define $S=\lim_{n\to \infty}\sum_0^na_n$,then, only convergent series can have meaningful $S$. But if we decide to define $S$, to be the Caesaro mean, now we can also assign a real ...
curiosity's user avatar
  • 151
0 votes
1 answer
25 views

How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary?

I'm looking for an example of a series of complex functions $\{f_n(z)\}$ with the following properties: The series $\sum_{n=1}^{\infty} f_n(z)$ converges uniformly on a closed region $D$ in the ...
John Title's user avatar
2 votes
1 answer
115 views

Limit $\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$ [duplicate]

I am currently trying to calculate the following limit of sequence: $$\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$$ I need it to prove that a series diverges, but I ...
Vito Palmieri's user avatar
0 votes
0 answers
58 views

How does one show that an alternating series $\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$ is unbounded/diverges? [duplicate]

I have arrived at the point where I have a series $$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$$ that I know should diverge (checking via python implies it might be divergent, and wolframalpha times out ...
artoftheblue's user avatar
1 vote
1 answer
51 views

Is it true that for any positive integer $n$, there exists an integer $x$ where there are at least $n$ primes between $x^2$ and $(x+1)^2$

Am I correct that this follows directly from two observations: (1) The sum of the reciprocals of primes diverges. (2) The sum of the reciprocals of squares converges Here's my thinking: If there ...
Larry Freeman's user avatar
-2 votes
1 answer
43 views

Find the total of a series, $\sum_{n=1}^{\infty} xc(1+c)^{n-1}$ only up to a specific number of decimal places.

Up-front: Met the snobs on Math overflow and thought I'd try here instead. I have three STEM degrees including math, and this is for work not school; just trying to save hours googling. This equation ...
Lance Roberts's user avatar
1 vote
1 answer
47 views

Regularization of $\lim_{n\to\infty}n^s$ using Zeta Function

I will be considering the Riemann Zeta Function to assign values to some limits. Values of $\zeta(s)$ at negative integer arguments are known. For example, $$\zeta(0)=-\frac{1}{2}\stackrel{\Re}{=}\...
Miracle Invoker's user avatar
1 vote
0 answers
53 views

Convergence of a finite series [closed]

How do I write a converged form for the following finite series? $\sum_{k=0}^{M-1}\frac{1}{k!}f(x)^k$ where M is any integer and x is variable If the convergence of this series is not possible, is it ...
kunal 's user avatar
  • 19
3 votes
2 answers
184 views

$f$ and $1/p$ are positive and decreasing, $\int_1^\infty f<\infty$, $\int_1^\infty\frac{1}{p}=\infty$, show $\lim_{x\rightarrow \infty}p(x)f(x)=0$

Here I'm interesting in prove (or disprove and counterexamples) the following Claim : Claim: For $p: [1,\infty)\longrightarrow\mathbb R$ and $f: [1,\infty)\longrightarrow\mathbb R$ where $p$ is ...
CCQ's user avatar
  • 127
0 votes
1 answer
93 views

Does this sequence of real number converge?

Given the sequence of real numbers $$a_n = \left(\sum_{k=1}^n \frac{1}{\sqrt{1+k^2}}\right)- \ln(n+\sqrt{1+n^2}) , n\in \Bbb{N}$$ Test its convergence. What I did was to separate the sequence from the ...
Fatou Sall's user avatar
0 votes
0 answers
53 views

Does smoothing divergent series with cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...
not all wrong's user avatar
0 votes
2 answers
59 views

Comparison test example from Spivak's Calculus

As an example application of the comparison test, Spivak introduces the series $\displaystyle \sum_{n=1}^{\infty} \frac{n+1}{n^{2}+1}$. He says, "we would expect this series to diverge, since $\...
joeshiki's user avatar
0 votes
0 answers
72 views

Proof of non-convergence of sine series

2 By considering the identity $\cos[(2n-1)\alpha]-\cos[(2n+1)\alpha]\equiv2\sin\alpha\sin2n\alpha,$ show that if $\alpha$ is not an integer multiple of $\pi$ then $\sum_{n=1}^N \sin (2n\alpha) = \...
umr0hazar's user avatar
2 votes
0 answers
86 views

Using the Banach-Steinhaus Theorem to show the existence of a divergent Fourier series

The proof of the existence of a continuous function $f\in C(\Bbb T)$ with divergent Fourier series using the Banach-Steinhaus theorem is well-known. It uses the unboundedness of the Dirichlet kernel $...
stoic-santiago's user avatar
0 votes
0 answers
47 views

Proving that the sum of prime reciprocals diverges using Borel Canteloni, 2nd version

I'm looking on feedback on the following proof. Most importantly, did I make any fundamental errors in my reasoning? If not, how could I make the proof more professional, adding rigour and better ...
AndroidBeginner's user avatar
0 votes
2 answers
110 views

Convergence or divergence of $\sum_{n=2}^{\infty}\frac{\ln \frac{n+1}{n}}{\ln \frac{n-1}{n}}$

I need to study convergence or divergence of this series $\sum_{n=2}^{\infty}\frac{\ln \frac{n+1}{n}}{\ln \frac{n-1}{n}}$. All the terms are negative hence $\frac{n-1}{n}<1$ which implies that $\ln ...
weymar andres's user avatar
-1 votes
1 answer
76 views

Spivak Ed.1 Chapter 22 Problem 1 (xi) [closed]

The problem asks to determine whether $$ \sum_{n=2}^\infty \frac{1}{(\log \ n)^n} $$ converges or not. The answer states that it converges because $ \frac{1}{(\log \ n)^n} \lt \frac{1}{2^n} \text{ for ...
MSU's user avatar
  • 185
0 votes
0 answers
34 views

Spivak ed.1 Chapter 22 problem 1 (v)

The problem asks to determine whether the following series is convergent of not: $$ \sum_{n=2}^\infty \frac{1}{\sqrt[3]{n^2-1}} $$ The solution states that it is convergent because for very big n ...
MSU's user avatar
  • 185
0 votes
1 answer
75 views

Regularizing a divergent sum

I have a sum of an infinite series $$ S = \frac{1}{3} - 4 + \frac{196}{15} - 21 + 27 - 33 + 39 - 45 + 51 - 57 + 63 + ... $$ which appears to diverge. This can be separated as such $$ S = (\frac{1}{3} ...
user8675309's user avatar
0 votes
1 answer
57 views

Determine $f_k(n)$ in $\sum_{i=0}^n \left (1-\frac{1}{2^i} \right )^k - f_k(n) \rightarrow 0$

For a given $k \in \mathbb{N}$, could you determine $f_k(n)$ such that the following holds? $$\sum_{i=0}^n \left (1-\frac{1}{2^i} \right )^k - f_k(n) \rightarrow 0$$ For $k=1$, $$ \sum_{i=0}^n \left (...
Amir's user avatar
  • 8,415
1 vote
1 answer
93 views

How can I show that the given series diverges? [closed]

How can I show that this series diverges? $$ \sum_{n=1}^{\infty} \frac{\bigl(2 + (-1)^n\bigr)^n}{\sqrt{n}\,3^n} $$ The standard tests kinda fail in this case. I am not getting any clue on how to ...
Confusedphysica's user avatar
0 votes
2 answers
83 views

Asymptotic evaluation of a series

I am reading a paper in which the authors derive the asymptotics for the following series: $$ \sum_{k=0;\,k+=2}^{N(1+m)-2}\left(1-\frac{k}{2}\log{\frac{k+2}{k}}\right)\sim \frac{1}{2}\log{\left(\pi N(...
Ruth Murphy's user avatar
0 votes
0 answers
33 views

If $\sum\frac{1}{a_n}$ diverges, then $\sum\frac{1}{n\max_{1\leq k\leq n}\frac{a_{n+1} - a_k}{n-k+1}}$ diverges.

Let $(a_n)$ be a strictly increasing sequence of positive real numbers, and denote $\Delta a_n:= a_{n+1} - a_n.$ We know that if $\displaystyle\sum_{n\in \mathbb{N}} \frac{1}{a_n}$ diverges, then $\...
Adam Rubinson's user avatar
2 votes
0 answers
62 views

Is my proof of the divergence of prime reciprocals valid

I tried to prove the divergence of the prime reciprocals as a challenge and I think I came up with quite an intuitive argument using Borell Cantelli, but maybe not rigorous. For two primes $p_n>...
AndroidBeginner's user avatar
2 votes
1 answer
87 views

Proofs involving manipulation of divergent series

Is this proof valid even though the harmonic series it is based on is divergent? Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$ Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
Older Amateur's user avatar
1 vote
1 answer
59 views

Numerically compute an oscillating series

I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
knuth's user avatar
  • 31
1 vote
0 answers
28 views

Convergence of derived series

I found this question on another forum so I don't have more details about it other than it seems non-intuitive to me: Let $\sum_{n=1}^\infty a_n$ consist of positive terms and be convergent. Define $...
Ivan's user avatar
  • 2,386
1 vote
1 answer
80 views

Does this imaginary series really diverge? [duplicate]

$$\sum_{n=2}^{\infty}{\frac{(-i)^{n}}{\ln n}}$$ In the answer, using comparison test $1/\ln n > 1/n,$ the series is divergent. But, in my opinion, the series can be divided like $$\sum_{n=1}^{\...
J sw's user avatar
  • 21
-2 votes
1 answer
104 views

How to evaluate $\sum \frac{1}{2^{n-r}}\frac{1}{n}$? [closed]

$$\sum_{n=r+1}^{\infty}\frac{1}{2^{n-r}}\frac{1}{n}$$ ,where r is a positive integer. First, I wanna know if this is convergent(I guess so) and WHY. And if so, how to evaluate it?
David Lee's user avatar
  • 185
0 votes
0 answers
10 views

Concoct an alternating series which diverges that is alternating, the limit goes to 0, but is NOT always decreasing. [duplicate]

Basically need an example of a divergent series that is alternating and the limit goes to 0.
Super walrus's user avatar
4 votes
1 answer
85 views

Divergence of Mehler's Hermite polynomial series

According to Mehler's formula, $$ \sum\limits_{n=0}^{\infty}\left(\cfrac{\rho}{2}\right)^n\cfrac{H_n(x)H_n(y)}{n!}=\cfrac{1}{\sqrt{1-\rho^2}}\exp\left(-\cfrac{\rho^2(x^2+y^2)-2\rho x y}{1-\rho^2}\...
george_ch's user avatar
1 vote
0 answers
67 views

Can a function be represented as 2 different power series

Find the power series representation for 𝑓(𝑥)=1/x+2 cant i write the power series as $$ f(x) = \frac{1}{x+2}. $$ so I am a little confused because cant I write the series as $$ f(x) = \frac{1}{1-(-x-...
barış yaycı's user avatar
0 votes
1 answer
53 views

Question about convergence and divergence of positive series,which means there exists no slowest convergent series. [closed]

If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence positive series $\sum_{n=1}^{\infty} b_n$,which satisfies $\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$ If ...
Dropsy Zheng's user avatar

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