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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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Rate of divergence of finite sum

What is the rate of divergence of $$f(n) = \sum_{k=1}^n \frac{1}{k^\alpha}$$ when $0 < \alpha <1$? I know that $f(n)$ diverges as fast as $ln(n)$ when $ \alpha =1$. I'm wondering whether ...
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14 views

comparing two decaying sequences

I have two sequences: $p_0, p_0 \rho_1, p_0 \rho_2, ...$ and $q_0, q_0 \gamma_1, q_0 \gamma_2, ...$. Both sequences have infinite number of terms, and both sequences sum to 1 each. All terms ...
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1answer
22 views

Contradiction in radii of convergence? Where is my error?

I'm working through Baby Rudin and I came across what seems to me to be a contradiction, but it could be an error on my part. It has to do with radii of convergence of power series. First, let $\{...
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2answers
63 views

How to convert the following sum to a geometric series?

Find $$ \sum_{n = 1}^{\infty} \frac{6 - 2^{2n - 1}}{3^n} $$ There are many ways to find that the limit is divergent, but the question explicitly states the sum must be interpreted as a geometric ...
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23 views

How to find if the Series diverges or not

I would like please to find if the series $\;\sum\limits_{n=3}^\infty\frac1{\sqrt n\log n}\;$ diverges or not. I tried to implement Cauchy's Criterio but it does not work. Thank you very much in ...
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1answer
35 views

Is the series $\sum_k\left| \frac{1}{\ln{(k+1)}}\right|^p$ divergent for $p > 1$

Let $p > 1$. Is the series $$\sum_k\left| \frac{1}{\ln{(k+1)}}\right|^p$$ divergent? This is in the context of finding a sequence that goes to zero but is not an element of $l^p$ space. I saw in ...
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2answers
258 views

Convergence/Divergence of infinite series

I'm trying to figure out if the following series converges or diverges. I have spent hours on it and can't figure it out. Tried to use the comparison test, Dirichlet, and Abel, but none of it worked. ...
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1answer
42 views

Prove that the logarithm family of sums diverges?

Define $$a_k(n) =\frac{1}{n\log(n)\log(\log(n)) \cdots \log^{k}(n)}$$ Do all of the sums (for a fixed $k \in \mathbb{N}$): $$A_k = \sum_{n} a_k(n)$$ Diverge?
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24 views

What does it mean when a series expansion has increasing negative order in epsilon as epsilon approaches 0?

I'm trying to get an order bound on the following integral as $\epsilon \to 0$: $$g(\epsilon) = \int_a^{a+\epsilon} f(\epsilon,v) dv$$ where $f$ is an ugly, but smooth, function when $a\leq |v| <a+...
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1answer
42 views

Divergence for Series with $|a_n| \ge b_n \ge 0$

first of all i hope you get my point since English is not my native. I know the series $$ y = \sum_{k=1}^\infty \frac{1}{k} $$ is divergent. Now I want to check if the series $$x=\sum_{k=1}^\infty ...
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5answers
148 views

$\sum_{n=1}^\infty \frac{n!e^n}{n^{n+ \frac{3}{2}}}$ - any ideas for a simple proof of divergence?

I am looking for a simple proof of divergence for the series: $\sum_{n=1}^\infty \frac{n!e^n}{n^{n+\frac{3}{2}}}$ That's a part of the more general problem: For what values of X is the series $\sum_{...
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1answer
29 views

Divergent condition of a series.

For $s>0$ when the following series will diverge $$\sum_{k=1}\bigg(\frac{2^k}{1+2^k}\bigg)^{2s}.$$
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22 views

Similar approach at series and integrals and d'Alembert's test.

We have "duality" between series and improper integrals. For example: We know that convergence doesn't depend on "proper part"*: If $\int_a^{\infty} f$ convergent and $b>a \to \int_a^{\infty} f = ...
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1answer
98 views

Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence?

$$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number. Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus ...
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3answers
67 views

Does $\sum q_i = \infty$ imply $\sum \log(1+q_i)=\infty$?

My intuition is that the first-order term in the Taylor expansion should dominate the series, if divergent: $$ \log(1+x) = x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 }...
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1answer
45 views

Improper integral convergence $I = \int_{0^+}^{1^-}\frac{\log(x)}{1-x}dx$

I was solving a few problems regarding convergence and divergence when I ran into this one. I tried searching on the internet but couldn't find an exact match to the problem. The task is to determine ...
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0answers
25 views

Regularization of the following sum

Am interested in how one could regularize the following sum $\sum_{m,n = 1, \infty} \sqrt{m^2 + n^2}$. Would preferably want this in the $\epsilon$ expansion regularization as talked below where a ...
2
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1answer
24 views

Divergent and convergent series of positive terms

I recently read an article which contains the following facts :--- Let $\{a_n\}$ be a sequence of positive number such that $\sum_{n=1}^{\infty} a_n$ diverges, so we must have $a_n\sim \frac{1}{n^p}$ ...
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2answers
275 views

Does $(1+\frac12-\frac13) + (\frac14+\frac15-\frac16)+(\frac17+\frac18-\frac19)+\cdots$ converge?

Does the series $$S=\left(1+\frac{1}{2}-\frac{1}{3} \right) + \left(\frac{1}{4}+\frac{1}{5}-\frac{1}{6} \right)+\left(\frac{1}{7}+\frac{1}{8}-\frac{1}{9}\right)+\cdots$$ converge? Here's my attempt ...
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2answers
32 views

Find radius of convergence of the power series.

Find the radius of convergence of power series $$ \sum_{n=0}^{\infty} 2^{2n} x^{n^2}$$ A)1 B)2 C) 4 D)1/4 I try to apply ratio and root test ( Cauchy–Hadamard theorem ) .but they ...
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1answer
41 views

How to tell if series terminates (Legendre ODE)

when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation $$a_{k+2}=\frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$ What I do not ...
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3answers
77 views

If $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty}\frac{1}{a_n}$ diverge?

If a series $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty} \dfrac{1}{a_n}$ diverge to infinity?
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0answers
57 views

Equality of perturbed Ramanujan's sum and -1/12

Consider the series $\sum_{k=1}^n k e^{-\varepsilon k}\cos(\varepsilon k)$. If one lets $\varepsilon$ be small enough (for example 0.0001) and $n$ large enough (for example 1.000.000), one sees by ...
2
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0answers
43 views

Closed form expression for (periodic) generalized harmonic numbers?

As far as I understand, there does not exist a pure closed form expression for the generalized harmonic numbers $H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}$ with $m\in\mathbb R$. My question is, however, ...
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2answers
46 views

Radius of convergence for three series

I need to find the radius of convergence of the 3 following series, but there are no solutions, so I don't know if my steps are correct. 1) $\sum_{n=0}^{\infty}x^{n!}$ 2) $\sum_{n=0}^{\infty} \frac{...
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1answer
23 views

Indefinite Sum Extension of a Finite Sum Equality

The other night I was considering the way in which we can split a finite sum of any arithmetic function into two finite sums, one for it's odd and another for even index terms : $$\sum _{k=1}^{n} \...
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7answers
123 views

Does $\sum\limits_{n=0}^\infty \frac{e^n\sin n}{n}$ converge or diverge? [closed]

The comparison test does not work on this, so I'm stuck trying to find a way to prove that it diverges. I know it definitely diverges. Any solutions?
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2answers
581 views

How does this series diverge by limit comparison test?

How does this series diverge by limit comparison test? $$\sum_{n=1}^\infty \sqrt{\frac{n+4}{n^4+4}}$$ I origionally tried using $\frac{1}{n^2}$ for the comparison, but I'm pretty sure it has to be $\...
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1answer
87 views

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
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2answers
69 views

Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?

I'm having trouble figuring out if the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$ Here's my thinking: $$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$ Since $\sum_{n=...
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0answers
22 views

Is the (x-a) format necessary when finding the radius of convergence for a geometric series?

So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments: Here is the work I did to ...
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3answers
53 views

Does the series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$ converge?

I have to find out if the series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$$ converges. Root test and ratio test did not work out for me. I also tried the alternating series test, but I can ...
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2answers
67 views

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge?

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge? I know that it converges, I just want to know what i'm doing wrong here. Here's what I did. $\sum\frac{-1}{\sqrt n} < $ $\sum \...
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0answers
35 views

This semi-harmonic-series converges [duplicate]

We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges. The series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\...
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2answers
19 views

Converging series and converging alternative series implies absolute convergence?

It is known that that harmonic series diverges, but the alternating form of the harmonic series converges. However, I am not sure if there are examples of series $a_n$ that $$ \sum_{n=1}^{\infty} ...
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0answers
89 views

Why do the Borwein integrals stop being $\frac{\pi}{2}$?

I just received the book "single digits - In praise of Small Numbers" by Marc Chamberland. In this book, he gives an interesting integral $$\displaystyle \int_0^\infty \dfrac{\sin x}{x} = \dfrac{\pi}...
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1answer
23 views

Radius of Finite Convergence

I am trying to solve a problem where I have to find the radius of finite convergence problem. I believe that I solved the problem correctly, receiving an answer of 1. However, I was informed that this ...
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1answer
69 views

Alternating Series Doubts

I've some doubts on Alternating series and Alternating test series. I was trying to clear my mind by practicing with some of my book exercises and found troubles with this one: $$\sum_{n=0}^\infty \...
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49 views

Applications/examples of these properties?

Here are two interesting properties on series : The first one : Let $(u_n)\in(\mathbb{R}_+)^{\mathbb{N}}$ such that $\sum \limits_{n\ge 0} u_n=+\infty$. Then there exists $(v_n)\in(\mathbb{R}_+)^{\...
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2answers
45 views

Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test

Image of My Work I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks P.S. if my ...
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1answer
33 views

How to prove when this series of polynomials is convergent and when it is divergent?

Let $P$ and $Q$ be polynomials of degree $k$ and $m$ and suppose $Q(n)\ne0\forall n\in\mathbb N$. Prove that $\sum_{n=1}^\infty\frac{P(n)}{Q(n)}$ is convergent if $m\ge k+2$ and divergent if $m\le k+...
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1answer
39 views

Determine whether the series is divergent or convergent and find its sum.

Any tips on how to start this would be great. I'm aware its not geometric but the answer key indicates it diverges, however I have no idea how to physically show that with this equation. $$\sum_{n=0}^\...
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3answers
50 views

Prove that a sequence is divergent (By definition - Epsilon-N Way)

First, this is the question: Prove (using epsilon-N definition) that the sequence $ a_n = \left<\sqrt{n}\right> $ is divergent. Note: $ \left<x\right> = x- \lfloor x \rfloor$ My ...
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1answer
35 views

If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $\frac1{n(1)}+\frac1{n(2)}+\ldots$ converge? [duplicate]

If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $$\frac1{n(1)}+\frac1{n(2)}+\ldots+\frac1{n(k)}+\ldots$$ converge? Actually, I have tried of ...
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1answer
40 views

Does this series diverge or converge?

Does this serie diverge or converge ? Why ? $$\sum \frac{\sqrt{n^3}+e^{-n}}{\sqrt{n^5}+\pi}$$
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1answer
49 views

Is analytic continuation well-defined as a summation method?

I am not well versed in summation methods or complex analysis, so I will be presenting a detailed view of my question with examples to illustrate my point as well as a few guiding questions that got ...
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1answer
28 views

Convergence of a series involving trigonometric ratios

How do I go about investigating the series $$ \sum _{n=1}^{\infty }\left(1+\frac{2n^{2}+n}{3n^3-n\sin (n)}\right)^{\tfrac{n^2+n\cos(n)}{n+\sin(n)}} $$ for convergence or divergence? What I don't ...
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1answer
63 views

(Proof Verification) Examining whether the series $\sum \limits_{n=0}^\infty \dfrac{(-1)^{n+1}}{5n+1}$ is convergent, absolute convergent or divergent

Everything in red is edited To show, that the series is convergent we show at first, that $\color{red}{\lim \limits_{n \to \infty} \left(\dfrac{1}{5n+1}\right)}=0$. $\color{red}{\lim \limits_{n \to \...
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1answer
21 views

How to determine whether the following two infinite series converge absolutely, converge conditionally, or diverge. [closed]

I need some guidance on how to solve these, I'm not understanding series and sequences too well and I need an explanation that hasn't come from my lecturer. $$\sum_{k=1}^\infty \frac{\log k}{k^2}$$ $...
2
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4answers
80 views

I was told that I couldn't “pull the limit in”. Tell me exactly how I'm messing up, please!

So, the problem that we were solving was $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ To figure out whether the series converged or diverged, after simplification, I asked my professor whether ...