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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1
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1answer
60 views

what is the value of $\zeta_{\mathbb{Q}(i)}(-1)$?

We've been told over and over $\boxed{\zeta(-1) = 1 + 2 +3 + 4 + \dots = - \frac{1}{2}}$ can be do the same over number fields? What should be the reasonable value for the zeta function $F = \mathbb{...
0
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5answers
63 views

Does this expression diverge or converge?

I have the following expression: $$\lim_{n \to \infty} \frac{2}{n^2} \ {\sum_{i=1}^{n}{\sqrt{n^2 - i^2}}} \ $$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to ...
1
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3answers
83 views

Will the expression $\sum_{i=1}^{n}{\frac{i^{2}}{n^{2}}}$ converge as n approches infinity?

I have the following expression: $$\lim_{n \to\infty}\ \sum_{i=1}^{n}{(\frac{i}{n})^{2}}$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to figure it out?
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1answer
47 views

1+2+3+4+5+… [duplicate]

How to visualize the divergent sum of 1+2+3+4+5+......=-1/12 Does it really exists?If it does how to visualize it
0
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0answers
13 views

Implications Divergence of one subseries and convergence of one subseries on convergence or divergence of series

If $\sum a_n$ is infinite series such that $\sum\limits_{a_n \geq0} a_n$ is diverging to $+\infty$ and $\sum\limits_{a_n \leq0} a_n$ convergent, can we say $\sum a_n$ and any other rearrangement of ...
2
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2answers
36 views

Sum of a series (combining divergent ones)

I am reading in a book, without any explanation, the following identity (with $a<b$): $$\sum_{k=0}^{\infty}\left(\frac{1}{k+a+1}-\frac{1}{k+b+1}\right)=\frac{1}{a+1}+\dots +\frac{1}{b}$$ ...
-2
votes
2answers
38 views

what is asymptotic behavior of $\sum \frac {1}{\sqrt[\alpha] k}$ [duplicate]

Asymptotic behavior of $$\sum \frac {1}{\sqrt[\alpha] k}$$ for $\alpha=1$? is $\ln k$ what about $\alpha > 1$ ? the suggested link is for $\alpha > \frac{1}{2}$ my question is about $ 0< \...
12
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1answer
257 views

Can different choices of regulator assign different values to the same divergent series?

Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = ...
-2
votes
1answer
18 views

$((n-1)^{0.5})/(((n+1)^2)-1)$ Is the sum convergent?, why or why not? [closed]

$$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks.
0
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0answers
36 views

Is this serie divergent? Calculate Max and Min $\sum_{k=1}^{\infty}ke^k$

This is the series: $$\sum_{k=1}^{\infty}ke^k$$ I need to calculate Max and Min with integral comparison. $$\int_1^nxe^x dx \leq \sum_{k=1}^n ke^k \leq \int_1^{n+1}xe^xdx$$ Calculate the indefinite ...
4
votes
4answers
206 views

How to prove that the series $\sum\limits_{n=1}^\infty {\sin^2n} $ diverges

I want to use a divergence test to prove that $\lim_{n\to \infty} \sin^2n$ does not converge. So $\sum_{i=1}^\infty \sin^2 n $ diverge. But because $\pi$ is an irrational number. So I cannot use ...
12
votes
1answer
117 views

The divergent sum of alternating factorials

So I came across this exposition of a paper by Euler here where Euler is trying to sum the divergent sum: $$s = 1 - 1 + 2! - 3! + 4! \dots = \sum_{k\geq 0}(-1)^k k!.$$ There are a couple of questions ...
2
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1answer
44 views

Different tests of convergence

For a non-negative sequence $(x_n)_{n\in\mathbb N}$, consider the following two criteria: $$\sum_{n=1}^\infty \frac{(\log n)^2}{x_n^2} < \infty \tag{1}\label{1}$$ and $$\exists r>1 \colon \sum_{...
2
votes
2answers
36 views

Comparision test for this series?

How do I check divergence of this series? $$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$ Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ...
0
votes
2answers
37 views

The sum of the perimeter of regular polygons inscribed inside of regular polygons

This is a question combining number theory and geometry. I am asking it purely from curiosity, but I think it might be a useful and interesting question. Start with an equilateral triangle of ...
2
votes
1answer
91 views

Ramanujan's divergent series

I tried to prove this sum by myself, but I couldn't. $1 + 4 + 9 + 16 + ... = 0$ First, I know this sums are a bit problematic, as we can't just $'='$ an infinite sum, but I would like to see the ...
1
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1answer
25 views

Do these series vanish asymptotically?

Let's consider a monotone increasing sequence $(K_N) \subseteq \mathbb{N}$ with $(K_N) \xrightarrow[N]{} \infty$ and $(K_N) = \mathrm o(N)$ (less increasing than $(N)$). Question: $\sum_{j=K_N + 1}^...
1
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2answers
64 views

Is this series divergent or convergent?

Please explain what method you used to prove so. $$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$
5
votes
1answer
133 views

Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?

$$\sum_{n=2}^\infty \frac{1}{\ln(n!)}$$ I tried by comparing it to $\sum_{n=1}^\infty \frac{1}{n}$ but i seem to fail. I think I need to compare with series that are smaller and diverge. Help.
2
votes
3answers
106 views

Divergence of $\sum\limits_{n=0}^{\infty} \dfrac{n^n}{n!e^n}$: fast proof?

The idea is to show that $\sum\limits_{n=0}^{\infty} \dfrac{n^n}{n!e^n}$ diverges, but that $\lim_{n \to\infty}\dfrac{n^n}{n!e^n} = 0$ (which is the reason why the series is challenging). I was ...
1
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0answers
43 views

Mobius function related series and Dirichlet summation

A well known identities of rare beauty is that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac 1{\zeta(s)}$$ Where $\mu(n)$ is the Mobius function and $\zeta(s)$ is the Riemann Zeta function. So, in ...
3
votes
3answers
148 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
85 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
94 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
93 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 evaluate....
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2answers
43 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
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3answers
48 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
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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
17 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
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0answers
38 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
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4answers
32 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
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1answer
26 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
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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 $\zeta(s)...
2
votes
1answer
31 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
70 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
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3answers
38 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
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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
27 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 ...
4
votes
0answers
127 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
68 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
67 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
28 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
51 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
45 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
71 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
90 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? $$...