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.
-1
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0answers
29 views
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 ...
0
votes
0answers
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 ...
2
votes
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 $\{...
1
vote
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 ...
-1
votes
0answers
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 ...
1
vote
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 ...
4
votes
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. ...
1
vote
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?
0
votes
0answers
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+...
0
votes
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 $$x=\sum_{k=1}^\infty ...
3
votes
5answers
148 views
$\sum_{n=1}^\infty \frac{n!e^n}{n^{n+ \frac{3}{2}}}$ - any ideas for a simple proof of divergence?
I am looking for a simple proof of divergence for the series: $\sum_{n=1}^\infty \frac{n!e^n}{n^{n+\frac{3}{2}}}$
That's a part of the more general problem:
For what values of X is the series $\sum_{...
-2
votes
1answer
29 views
Divergent condition of a series.
For $s>0$ when the following series will diverge $$\sum_{k=1}\bigg(\frac{2^k}{1+2^k}\bigg)^{2s}.$$
0
votes
0answers
22 views
Similar approach at series and integrals and d'Alembert's test.
We have "duality" between series and improper integrals. For example:
We know that convergence doesn't depend on "proper part"*:
If $\int_a^{\infty} f$ convergent and $b>a \to \int_a^{\infty} f = ...
1
vote
1answer
98 views
Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence?
$$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$
Where $e$ is Euler's number.
Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus ...
2
votes
3answers
67 views
Does $\sum q_i = \infty$ imply $\sum \log(1+q_i)=\infty$?
My intuition is that the first-order term in the Taylor expansion should dominate the series, if divergent:
$$
\log(1+x) = x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 }...
1
vote
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 ...
0
votes
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
votes
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 $a_n\sim \frac{1}{n^p}$ ...
5
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
-2
votes
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?
0
votes
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
votes
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, ...
0
votes
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{...
0
votes
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 :
$$\sum _{k=1}^{n} \...
1
vote
7answers
123 views
Does $\sum\limits_{n=0}^\infty \frac{e^n\sin n}{n}$ converge or diverge? [closed]
The comparison test does not work on this, so I'm stuck trying to find a way to prove that it diverges. I know it definitely diverges. Any solutions?
3
votes
2answers
581 views
How does this series diverge by limit comparison test?
How does this series diverge by limit comparison test? $$\sum_{n=1}^\infty \sqrt{\frac{n+4}{n^4+4}}$$
I origionally tried using $\frac{1}{n^2}$ for the comparison, but I'm pretty sure it has to be $\...
2
votes
1answer
87 views
Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?
Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
6
votes
2answers
69 views
Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?
I'm having trouble figuring out if the following series converges or diverges.
$$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$
Here's my thinking:
$$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$
Since $\sum_{n=...
0
votes
0answers
22 views
Is the (x-a) format necessary when finding the radius of convergence for a geometric series?
So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments:
Here is the work I did to ...
1
vote
3answers
53 views
Does the series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$ converge?
I have to find out if the series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$$ converges. Root test and ratio test did not work out for me. I also tried the alternating series test, but I can ...
2
votes
2answers
67 views
Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge?
Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge?
I know that it converges, I just want to know what i'm doing wrong here. Here's what I did.
$\sum\frac{-1}{\sqrt n} < $ $\sum \...
0
votes
0answers
35 views
This semi-harmonic-series converges [duplicate]
We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges.
The series
$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\...
0
votes
2answers
19 views
Converging series and converging alternative series implies absolute convergence?
It is known that that harmonic series diverges, but the alternating form of the harmonic series converges. However, I am not sure if there are examples of series $a_n$ that
$$ \sum_{n=1}^{\infty} ...
6
votes
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
$$\displaystyle \int_0^\infty \dfrac{\sin x}{x} = \dfrac{\pi}...
0
votes
1answer
23 views
Radius of Finite Convergence
I am trying to solve a problem where I have to find the radius of finite convergence problem. I believe that I solved the problem correctly, receiving an answer of 1. However, I was informed that this ...
0
votes
1answer
69 views
Alternating Series Doubts
I've some doubts on Alternating series and Alternating test series. I was trying to clear my mind by practicing with some of my book exercises and found troubles with this one:
$$\sum_{n=0}^\infty \...
0
votes
0answers
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}_+)^{\...
0
votes
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 ...
0
votes
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+...
1
vote
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.
$$\sum_{n=0}^\...
1
vote
3answers
50 views
Prove that a sequence is divergent (By definition - Epsilon-N Way)
First, this is the question:
Prove (using epsilon-N definition) that the sequence $ a_n = \left<\sqrt{n}\right> $ is divergent.
Note: $ \left<x\right> = x- \lfloor x \rfloor$
My ...
1
vote
1answer
35 views
If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $\frac1{n(1)}+\frac1{n(2)}+\ldots$ converge? [duplicate]
If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $$\frac1{n(1)}+\frac1{n(2)}+\ldots+\frac1{n(k)}+\ldots$$ converge?
Actually, I have tried of ...
0
votes
1answer
40 views
Does this series diverge or converge?
Does this serie diverge or converge ? Why ?
$$\sum \frac{\sqrt{n^3}+e^{-n}}{\sqrt{n^5}+\pi}$$
1
vote
1answer
49 views
Is analytic continuation well-defined as a summation method?
I am not well versed in summation methods or complex analysis, so I will be presenting a detailed view of my question with examples to illustrate my point as well as a few guiding questions that got ...
0
votes
1answer
28 views
Convergence of a series involving trigonometric ratios
How do I go about investigating the series
$$
\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)}}
$$
for convergence or divergence?
What I don't ...
1
vote
1answer
63 views
(Proof Verification) Examining whether the series $\sum \limits_{n=0}^\infty \dfrac{(-1)^{n+1}}{5n+1}$ is convergent, absolute convergent or divergent
Everything in red is edited
To show, that the series is convergent we show at first, that $\color{red}{\lim \limits_{n \to \infty} \left(\dfrac{1}{5n+1}\right)}=0$.
$\color{red}{\lim \limits_{n \to \...
-1
votes
1answer
21 views
How to determine whether the following two infinite series converge absolutely, converge conditionally, or diverge. [closed]
I need some guidance on how to solve these, I'm not understanding series and sequences too well and I need an explanation that hasn't come from my lecturer.
$$\sum_{k=1}^\infty \frac{\log k}{k^2}$$
$...
2
votes
4answers
80 views
I was told that I couldn't “pull the limit in”. Tell me exactly how I'm messing up, please!
So, the problem that we were solving was
$$\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 ...
