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.

Filter by
Sorted by
Tagged with
-3 votes
1 answer
50 views

Test for the convergence of the series $1/5 + 2!/5^2 + 3!/5^3+\ldots $

The series is $1/5 + 2!/5^2 + 3!/5^3+\ldots $ It's $n$th term will be $n!/5^n $ The problem I'm facing that I'm getting infinity using D'Alembert's ratio test.
user avatar
-3 votes
1 answer
47 views

Series Test, Calculus 2 [closed]

I want to determine whether the series $$\sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))}$$ converges or diverges, and I don't know what test I should use. Thank you.
user avatar
0 votes
1 answer
29 views

Improper integral comparison test of $\int _0^1\frac{\cos\left(x\right)}{x^{1/3}}dx\:$

I'm trying to find the convergence/divergence of this integral but I can't seem to find an integral to compare to $$\int _0^1\frac{\cos\left(x\right)}{x^{1/3}}dx$$ I tried to compare to integral of $\...
user avatar
1 vote
2 answers
58 views

Convergence of a series induced from given two series

I could not get any counter example so I am asking this question. Given $\frac{1}{n}\sum_{k=0}^{n}ka_k\to 0$ and $\frac{1}{n}\sum_{k=0}^{n}kb_k\to 0$, as $n\to \infty$, where $a_k,b_k \in (0,1)$. Is ...
user avatar
  • 161
2 votes
1 answer
40 views

Am I correct about this series question?

The problem was as follows: Let $a_n$ be a non-negative series. Assuming $\sum_{n=1}^{\infty}a_n$ converges does a) $\sum_{n=1}^{\infty}\left(a_n\right)^{\frac{3}{2}}$ converge? b) $\sum_{n=1}^{\infty}...
user avatar
3 votes
2 answers
63 views

Does this series converge or not?

$$\sum _{k=2}^{\infty }\:\frac{1}{\sqrt{k}\left(\ln k\right)^{\ln k}}$$ I've tried limit comparison with $\frac{1}{\sqrt{n}}$ and $\frac{1}{n^2}$ and also Cauchy condensation test but nothing seems to ...
user avatar
5 votes
3 answers
108 views

How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
user avatar
  • 8,284
0 votes
0 answers
38 views

Does the series $\sum_{n=1}^{\infty}(n^\frac{1}{n}-1)$ diverge? [duplicate]

I came across the series $\sum_{n=1}^{\infty}(n^\frac{1}{n}-1)$ and I am not able to apply any of the convergence test here. I know that the series $\sum_{n=1}^{\infty}n^\frac{1}{n}$ and $\sum_{n=1}^{\...
user avatar
  • 101
0 votes
0 answers
49 views

Does the analytic continuation method for infinite series always work and is unique?

Sorry if this is worded poorly, but I don’t know enough in the subject to word it better: $$\sum_{n=0}^{\infty}n=-1/12$$ according to $$\zeta(-1)=\sum_{n=1}^{\infty}\frac{1}{n^s}=-1/12$$ Is there ...
user avatar
  • 355
2 votes
0 answers
47 views

Convergence of $\sum_{k=1}^\infty \frac{3k}{k^2+k} \sqrt\frac{\ln(k)}{k}$

I'm having a hard time doing this problem. As this is homework, I'd appreciate a guidance towards a solution, not a full answer. Thanks in advance. I need to prove that the series $$\sum_{k=1}^\infty \...
user avatar
  • 23
1 vote
0 answers
24 views

The Method used in the solution

$$ \sum_{k=1}^∞ \frac{(x^k)} {(k^2)} $$ The question is, Check if it is divergent. Solution: Step1: $$ R_1 = \frac{1}{lim \frac{k^2}{(k+1)^2}} =1 $$ can someone explain step 1.Which method is used ...
user avatar
0 votes
0 answers
66 views

How can I prove $\displaystyle\sum_{n=1}^{\infty} \frac {(-1)^n \sin^2(8n)}{n}$ diverges?

Again, the series is $$\sum_{n=1}^{\infty} \frac {(-1)^n \sin^2(8n)}{n}.$$ Wolfy says it converges, at first I thought it would converge, but after thought I think it diverges, mathway says it ...
user avatar
  • 350
0 votes
1 answer
29 views

If $\varepsilon_n \rightarrow 0 $ do we have that $\sum_{n<N} \varepsilon_n b^n = o (\sum_{n<N} b^n)$

If $\varepsilon_n \rightarrow 0$ and $b > 1$ do we have that $\sum_{n<N} \varepsilon_n b^n = o (\sum_{n<N} b^n)$ as $N \rightarrow +\infty$ ? I've tried an Abel transformation but I was not ...
user avatar
3 votes
1 answer
70 views

Convergence (or not) of $\sum_{n=1}^{\infty}\frac{1!2!\cdots n!}{n^n}$

The title says for itself: Does the series $\sum_{n=1}^{\infty}\frac{1!2!\cdots n!}{n^n}$ converges? So, as soon as I've saw the factorial, could not help but use the ratio test: \begin{align*} \lim \...
user avatar
1 vote
0 answers
40 views

Sum of Reciprocal of Repeated Products [closed]

For what real values of $c\in\mathbb{R}$ does the following series converge and diverge? $$ \sum_{k=1}^\infty\frac{1}{\prod\limits_{\substack{1\leq j\leq k \\ j\neq -c}}\left(1+\frac{c}{j}\right)} $$
user avatar
  • 1,155
1 vote
1 answer
42 views

Does this recurrence relation of exponentials diverge?

Does the following recurrence relation diverge? $$x_{k+1} = x_k - \frac{e^{-x_k}}{(1+e^{-x_k})^2}$$ where $x_0 <0$. I have plotted the first one million steps of the sequence using $x_0=-1$ ...
user avatar
0 votes
6 answers
61 views

How to check the convergence or divergence of this alternating series: $\sum_{n=1}^{\infty}\frac{(-1)^n}{1+\sqrt{n}}$?

Check the convergence or divergence of this alternating series: $$\sum_{n=1}^{\infty}\frac{(-1)^n}{1+\sqrt{n}}$$ My attempt: I know that $$\frac{1}{\sqrt n}>\frac{1}{n}\tag 1$$ we conclude that $\...
user avatar
2 votes
1 answer
122 views

Does the series $\sum_{n=1}^{\infty}\frac{\sin n}{n}(-1)^{n}$ converges or not?

From other questions we could know the series $\sum_{n=1}^{\infty}\frac{\sin n}{n}$ converges (using the Euler equaiton) and $\sum_{n=1}^{\infty}\frac{|\sin n|}{n}$ diverges (using the Euler–Maclaurin ...
user avatar
1 vote
0 answers
26 views

Error term of $( a_0+a_1x^{-1}+\mathcal O(x^{-2}))(b_0+b_1y^{-1}+\mathcal O(y^{-2}))$

Suppose $$ f(x)= a_0+a_1x^{-1}+\mathcal O(x^{-2}),\quad x\to\infty $$ and $$ g(y)= b_0+b_1y^{-1}+\mathcal O(y^{-2}),\quad y\to\infty. $$ How would we quote the error term of the product $f(x)g(y)$ for ...
user avatar
1 vote
3 answers
52 views

Absolute convergence and conditional convergence of the sum $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\cdot(1+\frac{1}{n})^n}{n}$

I am trying to solve an exercise from an old exam from my university which no solution is uploaded. Basically, I have to determine if the series: $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\cdot(1+\frac{1}{...
user avatar
0 votes
1 answer
76 views

Convergence $\ln\left(1+\frac{1}{n^2}\right)$ [duplicate]

I´m asking for some help, since I am stuck with a proof for convergence like follows: $$\sum\limits_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$$ I tried to separate it: $$\sum\limits_{n=1}^\infty\ln\...
user avatar
2 votes
0 answers
47 views

Interpreting a potential bias of the Mertens function

In this question of some years ago on MO, the presence of a negative bias for the Mertens function was hypothesized. A key point for such a problem is how the bias is defined. For example, if we focus ...
user avatar
  • 16.6k
2 votes
2 answers
109 views

Valid proof that Euler's Constant $\gamma$ is between $0$ and $1$?

I was wondering if there is any reasonable way/theory to do calculations with divergent limits of a sequence. I was trying to prove that Euler's constant $$\gamma = \displaystyle{\lim_{n \to \infty}} \...
user avatar
  • 109
2 votes
1 answer
96 views

Closed form for the sum of the series $\sum^\infty_{n=1} \left( {(-1)^n}\left( 1+n\ln(\frac{2n-1}{2n+1}) \right) \right)$

If $a>2$ then $\sum\limits^\infty_{n=1} \left( {(-1)^n}\left( 1+n\ln(\frac{an-1}{an+1}) \right) \right)$ diverges by divergent test. Does it converge if $a=2$? Is it possible to find an exact form ...
user avatar
  • 441
1 vote
0 answers
77 views

Convergence of series with logarithm upon polynomial as terms [duplicate]

Consider the following sequence $a_n=\dfrac{\ln n}{n^p}$ where $p>1$. I want to check for the convergence of the series of that sequence i.e. $\displaystyle \sum_{n=1}^{\infty}a_n$. I intuitively ...
user avatar
1 vote
0 answers
46 views

Question about convergence of a series.

First of all, Happy new year! Sorry for the non specific title, There was not enough space for a good description. I have been trying to find a example for the following function $f$: Does there ...
user avatar
  • 115
2 votes
1 answer
42 views

Continuation of an asymptotic series originally defined for $z>0$ to $z<0$

Consider the well-known asymptotic expansion of the $\Gamma%$-function: $$\Gamma(x)\sim\left(\frac{x}{e}\right)^x \sqrt{\frac{2 \pi }{x}}\left[1+\frac{1}{12 x}+\frac{1}{288 x^2}-\frac{139}{51840 x^3}-\...
user avatar
2 votes
1 answer
64 views

$\sum_{n=1}^{\infty} a_n$ is a series and {$s_n$} is the sequence of partial sums. If $s_n = \frac{n^2+1}{4n^2-3}$, find $\sum_{n=1}^{\infty} a_n$

Only idea I have is to use $a_n = s_n-s_{n-1}$, which would mean $a_n = \frac{n^2 + 1}{4n^2 - 3}-\frac{(n-1)^2 + 1}{4(n-1)^2 -3}$. But I'm not sure if I'm thinking in the right direction. Or would ...
user avatar
  • 63
4 votes
1 answer
102 views

Does the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converge or diverge? [duplicate]

As you've seen from the title, I'm wondering whether the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converges or diverges? I'm struggling! My first thought was to rewrite it like this to ...
user avatar
0 votes
1 answer
86 views

Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence.

Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence. My thoughts are sequences as : { 1 , 2, 2, 2, 2, 3, 2, 2, 2, 4, ...} or ...
user avatar
-3 votes
1 answer
114 views

1-1+1-1+... = k+1/2?

Let $$F(s) = \sum_{n=0}^{\infty} f_{n}(s)$$ be a complex analytical function defined by a series (not necessarily a power series) that absolutely converges on an open set $s \in U$. Assume that the ...
user avatar
  • 553
0 votes
1 answer
52 views

How to make this series converge?

Let $\lambda>0$ be fixed, and $a,b>0$ positive real numbers. We have a series which is defined as $$\sum_{n=0}^{\infty}\bigg(\prod_{j=1}^{n}\frac{1}{1+\lambda\frac{1}{a+b(j-1)}}\bigg).$$ Is ...
user avatar
  • 81
4 votes
1 answer
64 views

Do the elements of this series converge to zero?

Let $\{a_n\}\subset [0,\infty )$. It is well know that if $$\sum_{n \in \mathbb{N}}a_n < \infty$$ then we must have: $a_n \to 0$ as $n \to \infty$. Now suppose that $a_n=a_n(u)$ with $\{a_n\}\...
user avatar
0 votes
1 answer
37 views

How do I check whether the sequence converges uniformly?

The sequence is: $$f_n=x^n-x^{n+1}=x^n(1-x),\quad x\in[0,1]$$ I need to check whether it converges uniformly, but I'm having doubts about the answer for $x\in(0,1)$. Below is what I've tried. I apply ...
user avatar
0 votes
0 answers
22 views

The sum of the series(in functional series)

The task is to find the sum of the series: I can calculate the sum of an ordinary positive number series, but as I understand it, this series is not an ordinary positive number series. Tell me how to ...
user avatar
  • 93
0 votes
2 answers
68 views

Decide whether the sum is convergent or divergent

$$\sum^{\infty}_{n=0}\frac{1}{n!}(\frac{n}{e})^n$$ I have tried both ratio test and root test, both results in $1$, which cannot give any conclusion. Also the series itself goes to zero as $n$ goes to ...
user avatar
3 votes
1 answer
90 views

The harmonic Series sequence.

Well! I was going through harmonic series from mathworld.worlfram I found harmonic numbers are really tough to calculate I was scribbling and wrote this thing $$\sum_{k = 1}^{x}H_k= \sum_{k =1}^{x}\...
user avatar
1 vote
0 answers
44 views

Cancelling values in sum of an infinite series

Question: Find the sum of all integer values of c such that $x^2 +cx+\frac{1}{4}c$ has two real, distinct roots. The discriminant must be greater than zero for a quadratic to have two real, distinct, ...
user avatar
  • 793
0 votes
0 answers
79 views

Expansion of square root of (a-b)

I am interested in expanding: $$\sqrt{a-b}$$ Assuming $a>0$ and $b>0$, and that $a>b$. When I write this in Wolfram alpha, the expansion is as follows: $$\approx \sqrt{-b} +\frac{a}{2\sqrt{-b}...
user avatar
0 votes
0 answers
33 views

Proof of existence of a rearrangement that diverges to infinity for a conditionally convergent sieries

I came across the Wikipedia page on the Riemann series theorem here (section 4.2) which includes a proof that for a conditionally convergent sequence $a_n$ there exists a rearrangement such that $\...
user avatar
  • 35
0 votes
0 answers
29 views

Existence of a bijective function from the set of natural numbers to itself such that the following condition satisfy

Does there exist a bijective function $g:\mathbb{N}\to \mathbb{N}$ such that $\sum_{n=1}^{\infty}\frac{g(n)}{n^2}<\infty$ My approach Suppose it does then the sequence $(\frac{g(n)}{n^2})_{n\in \...
user avatar
  • 329
1 vote
1 answer
84 views

Show that $\sum_{n=3}^{\infty} \frac{1}{(\log \log n)^{\log \log n}}\quad $ diverges.

This problem comes from Apostol's Ex. 8.15 Show that $\sum_{n=3}^{\infty} \frac{1}{(\log \log n)^{\log \log n}} \;$ diverges. My attempt: Note $ \log \log n$ is increasing and positive, so the term ...
user avatar
  • 553
1 vote
1 answer
41 views

The radius of convergence of $\sum \left(\frac{z^2-1}{z^2+1}\right)^n$

I'm trying to find the radius of convergence of complex series $$S=\sum_0^\infty\left(\frac{z^2-1}{z^2+1}\right)^n$$ with the help of ratio test. With simple observation that the term $a_n$ become ...
user avatar
2 votes
1 answer
58 views

Test for Divergence Using Divergence Criterion for Functional Limits

Can I prove divergence of $\sum_{n=1}^\infty\frac{1}{\cos(n)+2}$ by showing that $\lim_{x\to \infty}\frac{1}{\cos(n)+2}$ does not exist? I know there is very simple way to solve it, but I am thinking ...
user avatar
0 votes
0 answers
36 views

Convergence/Divergence of series

I'm stuck in a problem "Using Limit Comparison Test determine for which values of p the series converges or diverges to" $$ \sum_{k=1}^\infty\frac{1}{k^p*\log_{10}k}$$
user avatar
-3 votes
2 answers
118 views

Is the series expansion of $e^x$, $\sin x$ & $\cos x$ valid for any real $x$? [closed]

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots$$ $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$ Are the above valid for any real $x$?
user avatar
1 vote
0 answers
56 views

convergence of a series by permuting its elements [duplicate]

Let $f: \mathbb{N} \to \mathbb{N} $ be a bijective function, the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {n} $ converges if and only if the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {...
user avatar
2 votes
2 answers
86 views

Prove that the series $\sum_{n=1}^{\infty} x_n$ diverges in a Hilbert space $~X$

Let $~X~$ be a Hilbert space and $~(e_n)_n~$ be an orthonormal basis for $~X.~$ Let us consider a sequence $~~(x_n)_n~~$ in $~~X~~$ given by $$x_n=e_n+\frac{e_n}{n+1},~~~~\text{ for all }~n \leq 1.$$ ...
user avatar
  • 1,551
2 votes
1 answer
140 views

In an alternating series remainder where the 1st term in remainder is a negative, why is the approximate series an overestimate?

Saw this answer but it doesn't go deep enough to help understanding this. I need help identifying my knowledge gap as I struggle to understand why it is an overestimate. My thinking below results in ...
user avatar
  • 643
3 votes
4 answers
100 views

Does $\sum_{n=1}^\infty \left(1+\frac{1}{n}\right)^{n^2}\left(\frac{1}{e}\right)^n$ converge?

$$\sum_{n=1}^\infty \left(1+\frac{1}{n}\right)^{n^2}\left(\frac{1}{e}\right)^n$$ With the Root Test I get: $e\times \frac{1}{e}=1$, which doesn't determine whether it converges or not. With the ...
user avatar
  • 593

1
2 3 4 5
33