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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3
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1answer
33 views

Does the series converge

We know that all series of the following form diverge: \begin{equation} S_k = \sum_{n=\left\lceil \mathrm{e}^k \right\rceil}^\infty \frac{1}{n (\ln n) (\ln \ln n)\dots(\ln^k n)} \end{equation} where ...
2
votes
2answers
36 views

Prove the divergent series $\sum_{k=2}^{\infty} \frac{1}{log^3k}$

Prove that the series $$\sum_{k=2}^{\infty} \frac{1}{log^3k}$$ diverges. I have already tried the ratio test and root test but both give me that it's less than 1, but when I wanted to check it on ...
0
votes
0answers
30 views

Good problems to do while reading Hardy's book on divergent series?

I am reading Hardy's text on divergent series and to my great dissapointment it has no exercises. I wonder if anybody among you know of some suitable references with problems to read simultaneously ...
2
votes
2answers
62 views

Convergence of $\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}}$

$$\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}} .$$ I believe the series diverges. I am thinking of using the integral test to show this, but I am not sure if that is right.
1
vote
2answers
37 views

Converging or Diverging Series

What test do i use to show this series converges or diverges? $$\sum_{r=1}^{\infty}\frac{1}{(1+\frac{1}{r})^{r}}$$ I know that $(1+\frac{1}{r})^{r} \rightarrow e$ so does this function converge to ...
1
vote
1answer
78 views

How does zeta of zero equal to negative one half rather than to infinity?

$$\zeta(0)=(1/1^0)+(1/2^0)+(1/3^0)+(1/4^0)+(1/5^0)...$$ Am I right? Anything raised to the power of zero is one. One to the power of zero is one. One divided by one is one. $$1/1^0=1$$ Am I right? ...
1
vote
1answer
35 views

Why are alternating divergent series generally easier to evaluate? [closed]

Why is it that alternating divergent series tend to be easier to evaluate or that there are more ways to evaluate them? Is there a particular reason for the difficulty to evaluate series that don't ...
0
votes
1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
-1
votes
1answer
32 views

A way to evaluate $\sum_{n=0}^\infty(-1)^{n+1}n$

The binomial expansion of $(a+b)^{-2}$ is given as $$(a+b)^{-2}=\sum_{n=1}^\infty(-1)^{n+1}na^{-1-n}b^{n-1}\tag{I think}$$ And when $a=b=1$, $$2^{-2}=\sum_{n=1}^\infty(-1)^{n+1}n=1-2+3-4+\dots$$ ...
1
vote
0answers
34 views

Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. ...
2
votes
1answer
25 views

About the comparison test

The comparison test for series tells us (quoted from wikipedia): "If the infinite series $\sum b_n$ converges and $0 \leq a_n \leq b_n$ for all sufficiently large $n$ (that is, for all $n>N$ for ...
2
votes
0answers
49 views

Regularizing the sum of all factorials

Consider the series $$\sum_{n=0}^\infty n! = 0! + 1! + 2! + 3! + 4! + \ldots = 1 + 1 + 2 + 6 + 24 + \ldots$$ This series clearly diverges. Now, given that the Gamma function is defined by $$n! = ...
2
votes
2answers
62 views

Does $\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$ converge/diverges??

Does this series converge or diverge? $$\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$$ my thought is that, I can break it down to $$\sum_{k=1}^{\infty}\ln(k) - \sum_{k=1}^{\infty}\ln(k+1)$$ then maybe ...
5
votes
2answers
78 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
0
votes
0answers
17 views

Generalisation of the sum operator for the divergent geometric series [duplicate]

I define a generalization of the sum operator as a linear operator from $C^N$ to C that matches with the already known operators (like the zeta regularization). With this technique, one can calculate ...
1
vote
1answer
31 views

Confusion when finding convergences using divergence and integral test?

I am having a bit of confusion doing the divergence and integral tests, specifically when I am trying to visualize the functions to get a better idea of why the methods work. For example, take the two ...
0
votes
2answers
49 views

Divergent bounded series such that limit of the difference between two trace elements is zero

I'm asked to give an example of a series $a_n$ which is bounded, have no limit and upholds this rule: $\lim_{n\to \infty}|a_{n+1}-a_n|=0$. I tried a lot of series but it didn't work.
-1
votes
3answers
44 views

Limit involving a diverging series [closed]

It is a known fact that the value of the sum $\Sigma \frac{1}{i}$ (when $i$ ranges over the integers) Diverges. But what is the solution of the following limit $$\lim_{n \to \infty} ...
2
votes
1answer
18 views

(Resolved) Does the sum of a subset of the Harmonic sequence converge iff its density approaches 0?

Update: This question has been resolved. I have made some mistakes in this post. I will leave my post here for readers to find out my mistakes. I have noticed that the post is a bit too long. So if ...
6
votes
1answer
43 views

Sum of the reciprocal of the prime-position primes.

The primes are $2, 3, 5, 7, 11, 13...$ The sum of the reciprocals of the primes diverges, proven by Euler: $$\sum_{n=1}^\infty{\frac{1}{p_n}}=\infty$$ Here, $p_n$ is the $n$-th prime. I'm asked to ...
0
votes
2answers
30 views

Limit divergence test

$$\sum_{n=1}^{\infty}\frac{3^n-1}{3^{n+1}}$$ The answer to this is that it diverges but I have no idea how to take the limit of this. It looks like if we direct sub in $n = \infty$ it will be ...
1
vote
2answers
45 views

Product of converging series

$\displaystyle \sum_{n=1}^∞ (-1)^n\dfrac{1}{n}.\dfrac{1}{2^n}$ Knowing that An alternating harmonic series is always convergent Riemann series are always convergent when $p>1$ Is it safe to ...
0
votes
2answers
32 views

relationship between convergence of a sequence and its corresponding series.

I'm preparing for my calculus exam and I'm unsure how to approach these type of questions . If the sequence $a_n$ is convergent/divergent what can we about the corresponding series $\sum_{n}a_n$? Is ...
2
votes
3answers
199 views

Divergent sum of factorials

Is it possible to get an exact value of the sum (using divergent series summation methods) $$ \sum_{n=0}^\infty~ \frac{(n+k)!}{n!} \quad?$$ where $k$ is a positive integer. The only other divergent ...
1
vote
1answer
16 views

How to find a sequence $\{b_n\}^{\infty}_{n=1}$based on the following assumption?

How to find a sequence $\{b_n\}^{\infty}_{n=1}$ such that $b_n\gt 0$ for all $n\ge 1$ and $\lim\limits_{x\to\infty}b_n=0$ and the series $\sum_1^\infty (-1)^{n+1}b_n$ is divergent. For ...
1
vote
2answers
46 views

Determining whether series diverges or converges

$$\sum_{n=1}^\infty \left(\frac{1-n}{2+3n} \right)^n$$ The question is to determine whether the series converges or diverges. I tried to distribute the power $n$ to the numerator and the denominator ...
1
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1answer
14 views

Are all series in the elementary Ramanujan class R = 1 non-summable by analytic continuation of Dirichlet series?

We say that a series $\sum_{n=1}^\infty a_n$ and the corresponding power series $f(x)=\sum_{n=1}^\infty a_nx^n$ belong to the Ramanujan class $R=1$ if $g(x)=f(x)-f(x^2)$ is Abel summable at $x=1$ ...
0
votes
1answer
42 views

Its the sum of the series $1/2+2/2+3/2+4/2… = -1/24$?

If $1+2+3+4... = -1/12$ then, $(1+2+3+4...)*1/2$ should equal $-1/24$ But I find this strange since the second infinite is larger than the first because $1/2+2/2+3/2+4/2\dots$ contains all integers ...
0
votes
1answer
24 views

Is there a term for adding a variable to an infinite series so that it turns from divergent to convergent?

I am looking for the term in which if you add a variable to a series, what is the first number in which that variable will be that would turn a divergent series to convergent. Example: If I take the ...
0
votes
1answer
28 views

Ambiguity in the basic comparison test.

I was given this problem: $\sum_{n=1}^\infty \frac{1}{n!} \\ \text{By the comparison test :} \\ \text{If} \sum_k^{\infty} a_n <= \sum_k^{\infty} b_n \text{ Then} \text{ if } b_n \text{ diverges ...
0
votes
2answers
39 views

How to prove the divergence of the following sum: $\sum^{\infty}_{x=1}\cot^{-1}\left[\frac{1}{1+x^2+x}\right]$

I'm having difficulty proving the divergence of the following infinite sum : [Note: Here, $\cot^{-1}$ is inverse $\cot$, i.e. arccot] $$\sum^{\infty}_{x=1}\cot^{-1}\left[\frac{1}{1+x^2+x}\right]$$ ...
0
votes
2answers
31 views

An alternating series convergence

I was wondering whether the following series converges or diverges, $$\sum_{n=1}^\infty (-1)^n \sqrt[n]{a}$$ $$\forall a>0, a\ne1$$ The divergence test cannot be applied, since the sequence does ...
1
vote
2answers
60 views

Followup-question: Does $ 1 + \rho /2 + \rho^2 /3 + 1/4 + \rho /5 + \rho^2 /6 +… $ with $\rho^3=1$ converge?

In the recent question it was asked whether $ 1 + 1/2 - 1 /3 + 1/4 + 1 /5 - 1 /6 +... $ converges and was answered to the negative. Just out of curiosity I looked numerically ...
0
votes
2answers
80 views

Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + …$ converge?

Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + ...$ converge? I know that $(a_n)= 1/n$ diverges, and $(a_n)= (-1)^n (1/n)$, converges, but given this pattern of a negative number every ...
1
vote
1answer
22 views

Conditions for Alternating series to diverge

Series $(-1)^nb_n$ converges conditionally if the series $\{b_n\}_{n=1}^{\infty}$ diverges but two conditions are satisfied: the series is non increasing . $\displaystyle\lim_{n\to \infty}{b_n} = ...
1
vote
2answers
46 views

Convergence or divergence of $\sum_{n=1}^\infty (-1)^{n+1}(2+(-1)^n)/n$? [duplicate]

Is $$\sum_{n=1}^\infty (-1)^{n+1}{(2+(-1)^n)\over n}$$ convergent or divergent?
0
votes
1answer
62 views

Rudin's Real and Complex Analysis Chapter 15 Q.11

I'm having a go at Rudin's Real and Complex Analysis Q15.11 (this is just for fun). Under what conditions on a sequence of real numbers $y_n$ does there exist a bounded holomorphic function in the ...
0
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0answers
19 views

Asymptotic Expansion for Function with an Embedded Integral [duplicate]

So I'm trying to find the asymptotic expansion as $x \to \infty$ of: $$f(x)=\frac{1}{\bigg[A-\int_{x_0}^x\frac{\lambda^y}{\Gamma(y+1)}dy\bigg]^{\frac{1}{\alpha}}}$$ where $x_0>0$ and $\alpha>0$ ...
1
vote
1answer
26 views

Evaluate if series with exponential diverges or converges

The task is to evaluate for what values of $a \in \Bbb R_+$ does the series $$\sum_{n=1}^\infty \frac{a^n \times n!}{n^n}$$ converge. I've already checked with the ratio test that it converges for $ a ...
2
votes
0answers
33 views

Why do the integral and the partial sum agree for small $a$ and $m$?

Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( ...
2
votes
1answer
55 views

If $\sum_{n=1}^\infty \ln(1+a_n)$ coverges, does $\sum_{n=1}^\infty a_n$ also converge?

If $\sum_{n=1}^\infty \ln(1+a_n)$ coverges and $a_n>0$ for all $n>0$, does $\sum_{n=1}^\infty a_n$ also converge? I know that the converse of this is true, but would this also be true?
0
votes
1answer
58 views

Determine if $\sum_{t=1}^\infty (-1)^{n+1}\frac{(-4)^n}{n4^n}$ converges or diverges.

Determine if $$\sum_{t=1}^\infty (-1)^{n+1}\frac{(-4)^n}{n4^n}$$ converges or diverges. To make it simpler to deal with, I managed to simplify the sum to $$\sum_{t=1}^\infty (-1)^{2n+1}\frac{1}{n}$$ ...
2
votes
1answer
43 views

Determine if $\sum_{t=0}^\infty \frac{(-1)^{t^2}t^2}{4+t^2}$ is absolutely convergent, conditionally convergent, or divergent

Determine if $$\sum_{t=0}^\infty \frac{(-1)^{t^2}t^2}{4+t^2}$$ is absolutely convergent, conditionally convergent, or divergent. I have tried the Ratio test, but the limit comes out to be $1$, which ...
2
votes
5answers
276 views

What is the importance of knowing if a series converges or diverges? [closed]

This semester I have been given a multitude of techniques for discovering if a series converges or diverges with no explanation for why I would need to know this. It would be helpful if I could better ...
2
votes
0answers
59 views

How to prove that the Euclidean norm of a vector valued sequence diverge to infinity?

Thanks for reading me. I have a very particular problem. I want to know whether the following is satisfied: $\lim_{k \rightarrow \infty} |x_k| = \lim_{k \rightarrow \infty} | (\sum_{j=0}^{k} (A + ...
3
votes
1answer
82 views

Series diverging faster than any polynomial

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence on $\mathbb{R}$, and let $p$ be a polynomial with positive leading coefficient. Let me state a definition first: Divergence of a sequence to $\pm\infty$ A ...
2
votes
3answers
93 views

Convergence of $\sum \limits_{n=1}^{\infty}(-1)^{n+1}\frac{n}{n^2+1}$ [closed]

Test the absolute convergence and convergence of the following series. $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n}{n^2+1}$$ Am unable to apply the alternating series test. The book answers it as ...
2
votes
1answer
46 views

Show divergence of infinite series $\sum \frac{1}{\sqrt{(n(n+1)}}$

I am having trouble showing that this series is divergent. I do see that $\frac{1}{\sqrt{(n(n+1)}} = \frac{1}{n\sqrt{1+ \frac{1}{n}}}$. However, I can't find a series that is smaller and diverges to ...
2
votes
1answer
48 views

Show that if $\sum b_n$ is a rearrangement of a series $\sum a_n$ , and $a_n$ diverges to $\infty$, then $\sum b_n = \infty$

I was presented with the following problem; Show that if $\sum b_n$ is a rearrangement of a series $\sum a_n$ , and $a_n$ diverges to $\infty$, then $\sum b_n = \infty$. How would one solve this? ...
1
vote
1answer
29 views

$ |a_{n+1}/a_n| \leq n^2/(n+1)^2 $ for all natual numbers $n$, prove that the series formed by $a_n$ converges absolutely

$ \left|\dfrac{a_{n+1}}{a_n}\right| \leq \dfrac{n^2}{(n+1)^2} $ We are given the above infinite series and are to prove its absolute convergence. Clearly, ratio test isn't of use here and I can't ...