Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

learn more… | top users | synonyms

2
votes
3answers
61 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
4
votes
3answers
71 views

What is $\lim_{x\to \infty} 2\sqrt{x}- \sum_{n=1}^x {1\over \sqrt{n}}$? [duplicate]

I ask this because I noticed the partial sum $\sum_{n=1}^x {1\over \sqrt{n}}$ is very close to $2\sqrt{x}$, so close in fact that it appears their difference approaches a constant value, like $H_x$ ...
5
votes
2answers
82 views

Is the series convergent

Is series $\sum_1^\infty \frac{\ln(1+1/2) \ln(1+1/4) \cdots \ln(1+1/(2n))}{\ln(1+1/1) \ln(1+1/3) \cdots \ln(1+1/(2n-1))} = \sum_{n=1}^\infty \prod_{m=1}^n \ln(1+1/(2m))/(\ln(1+1/(2m-1))$ convergent ?
2
votes
2answers
86 views

Decide whether the series ${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$ converges or diverges

Determine whether the series converges or diverges $${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$$ I was thinking I should use ratio test but I get an ugly sequence that I don't know how to ...
1
vote
2answers
40 views

How to prove $\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$ diverges?

$$\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$$ It seems clear to me that this series diverges because the dominant temr is $1/n^{2/3}$, a p-series with $p < 1$ However I need to prove ...
1
vote
2answers
36 views

How to prove $\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $ diverges?

$$\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $$ It seems clear to me that this seires diverges since the dominant term is $(3/2)^n$, a geometric series with $r > 1$ However I am required to ...
0
votes
0answers
36 views

Proof that $\sum_{j=0}^\infty C_j$ converges if $\sum_{j=0}^\infty \|C_j\|$ converges

$C_j$ is a sequence of matrices in $\mathbb C^{n \times n}$ and the identity $$\max_{j,k}|A_{j,k}|\leq \|A\|\leq n\max_{j,k}|A_{j,k}|$$ is known. Show that $\sum_{j=0}^\infty C_j$ converges if ...
0
votes
2answers
15 views

Convergence test from Demidovich

I'm just learning how to test series for convergence and have encountered this series from the Demidovich's book and I can't really decide what criteria should I use. Could you please give me some ...
0
votes
0answers
37 views

Why is the infinite series of 0^n divergent?

No lie, Wolfram Alpha just told me it is. My input was "sum from 1 to infinity of 0^n" and its output was "By the geometric series test, the series diverges", but without interpreting as a geometric ...
0
votes
4answers
29 views

Convergence when the comparison test cannot be applied

I had a standard problem in my textbook which was to determine the convergence of $\sum _{n=2}^\infty\frac{n^3+1}{n^4-1}$. To determine whether the series is convergent or not the standard solution ...
0
votes
1answer
24 views

what is the general way of showing a sequence diverges and how would you manipulate this method so it works for series?

How could we show the divergence of a sequence solely by using the definition of divergence ? Also how can this general method be manipulated to work for series too?
0
votes
1answer
62 views

Is the given expression convergent as $n\to\infty$?

I want to know whether the following expression is convergent as $n\to\infty$ $$\frac{1}{n}\sum\limits_{k=1}^{\infty}\frac{|\ln n-\ln k|}{k^{(1+1/n)}}\cdot$$ With use of Riemann zeta function ...
2
votes
1answer
28 views

Proving that an alternating sequence does not converge

I have the sequence $$(a_n)_{n \in {\mathbb{N}}} = \lim_{n\to\infty} \frac{\frac{n!}{n^n}+1}{\frac{n!}{n^n}+(-1)^n}$$ I can see intuitively why this doesn't converge as it acts like $(-1)^n$ for large ...
3
votes
0answers
68 views

Harmonic series derivate - convergent or not?

It is known that $\sum_{n=1}^\infty \frac{1}{n}=\infty$, $\sum_{n=1}^\infty \frac{1}{n\ln(n)}=\infty$, $\sum_{n=1}^\infty \frac{1}{n\ln(n)\ln(\ln(n))}=\infty$ etc. But what happens if we consider the ...
0
votes
1answer
29 views

Does the alternating series converge?

I'm trying to find out whether the series $$\sum\limits_{n=1}^{\infty}(-1)^n\ln\left[\frac{8n+2}{7n+1}\right]$$ converges or not, but the alternating series test seems not to apply. What other tests ...
-1
votes
3answers
37 views

Does the series diverge or converge and find the sum if possible. [closed]

Does the series diverge or converge? $$\sum_{n=1}^\infty \frac{3}{5^n - e^n}$$
1
vote
1answer
27 views

Existence (and construction) of a convergent series

Suppose $(b_n)$ is an unbounded (real or complex) sequence. Does there always exist some (absolutely) convergent series $\sum a_n$ such that $\sum |a_nb_n|$ (or better, $\sum a_nb_n$) diverges? If so, ...
4
votes
1answer
44 views

Prove if $4$ kinds of series either converge or diverge.

I summed up $4$ kinds of series that I am having trouble solving. It seems like for the first one, the limit in infinity is - infinity which means it diverges. The limit when $x \to \infty$ in 2 is ...
1
vote
1answer
25 views

divergence of $\sum_{n \in \mathbb N} \frac{|a_n|^2}{1 + \sum_{k=1}^n|a_k|^2}$ when $\sum_{k \in \mathbb N} |a_n|^2 = \infty$

How can I proof that the divergence of the series $$\sum_{n \in \mathbb N} \frac{|a_n|^2}{1 + \sum_{k=1}^n|a_k|^2}$$ when $\sum_{k \in \mathbb N} |a_n|^2 = \infty$ I've been trying Cauchy test, ratio ...
3
votes
0answers
103 views

Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not ...
2
votes
2answers
67 views

Smallest $x$ for which $\sum_{n=1}^{\infty}\dfrac{1}{n^x}$ converges

Consider the series $$S_x = \sum_{n=1}^{\infty}\dfrac{1}{n^x}$$ for $x > 0$. Then $S_1$ is the harmonic series, which is known to diverge. $S_2 = \dfrac{\pi^2}{6}$; this is the Basel problem ...
5
votes
2answers
66 views

Can we cover the entire plane with the square with area 1/n for each positive integer n?

We have one square with area 1/n for each positive integer n. Is it possible to place these squares in the xy-plane in such a way that they completely cover the entire plane. If Yes, can you describe ...
2
votes
0answers
25 views

Tricky divergent binomial expansions?

The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as $$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$ In some situations, we can find the result of a ...
4
votes
1answer
50 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
40 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
32 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
67 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
38 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
83 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
57 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
35 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
45 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
28 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 ...
3
votes
0answers
60 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
63 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
80 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
22 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
50 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
46 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
21 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
46 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
1answer
31 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
46 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
34 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
206 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
49 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
vote
1answer
15 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 $x=1$ ...