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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Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
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2answers
23 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
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4answers
70 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
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2answers
26 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
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1answer
27 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
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1answer
38 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
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3answers
60 views

How does this series diverge?

The series: $$\sum_{n=0}^{\infty} \sqrt{n^2 +1} -n$$ diverges. Can someone please tell me how this is proven and done.
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2answers
28 views

Determining Divergence

How can I prove that this series diverges? I don't think you can use a comparison test, but maybe I'm mistaken. $$\sum \dfrac 1{n^{4/5}+10^{10}}$$
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3answers
76 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
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4answers
131 views

Why do some series converge and others diverge? [closed]

Why do some series converge and others diverge? What causes the divergence or convergence of a series and why does that cause such a behavior? For example, why does the harmonic series diverge, but ...
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1answer
32 views

Convergence of series involving n!

Let $$a(n)=(n-1)!\frac{e^n}{n^{n-1/2}} - \sqrt{2\pi}$$ for n=1 to infinity. Does the sum of $a(n)$'s, i.e. $\sum_{n=1}^\infty a(n)$, converge?
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1answer
37 views

Divergence of $\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}…$

Assume that we know this converges. $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+....$$ Is it possible to detect for which largest $0<s<1$ the sum below diverges? ...
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2answers
34 views

Help with Convergence/Divergence

So I am trying to prove whether the following problem converges or diverges? $$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$ So I decided to use the Root test. $$ L = \lim_{n\to ...
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0answers
105 views

How find this series $\sum_{n=1}^{\infty}\frac{1}{n^2H_{n}}$?

Question: This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$ I suddenly thought of this ...
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0answers
54 views

Good book on analytic continuation?

For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include a chapter or two on ...
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2answers
99 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
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3answers
87 views

Divergent Alternating Series

Need help in finding an alternating series: S = $\sum_{n=1}^{\infty}(-1)^{n+1}b_n$ where $\lim_{n\to \infty}b_n = 0$ $b_n > 0$ but only $\forall n \ge 1$ such that S diverges
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1answer
35 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
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4answers
159 views

Find the regularized sum $\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+…$

By considering the integral Zeta function $$F(s)=s+\frac{1}{2^s\ln(2)}+\frac{1}{3^s\ln(3)}+\frac{1}{4^s\ln(4)}+...$$ Evaluate $$\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+...$$ EDIT: ...
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1answer
55 views

Sum of reciprocals of every nth prime

I'm looking for a proof that $\displaystyle\sum_{n\mathop=1}^{\infty}\frac{1}{p_{kn}}$ diverges, where $p_n$ denotes the $n$th prime number and $k$ is a natural number. I know the proof that ...
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2answers
38 views

Does it converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get 1. I suppose I should compare it with some other series, but I can't figure out with ...
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1answer
46 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
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2answers
53 views

Determine whether the following series convergent? [closed]

Is the following series convergent? $$\sum_{k = 1}^{\infty}{1 \over k\left[1 + \ln^{2}\left(k\right)\right]}$$
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4answers
268 views

Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$
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1answer
274 views

The series $a_n$ is conditionally convergent. Prove that the series $n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0 $ ...
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0answers
41 views

Is there an infinite hierarchy of sequences whose reciprocals diverge, starting with the natural numbers?

It is well known that the sum of the reciprocals of the function $f_0(n)=n$ (the harmonic series) diverges: $$\sum_{n=1}^\infty\,\frac{1}{n}=\infty$$ Similarly, the sum of the reciprocals of the ...
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0answers
31 views

Naive calculations with infinite series [duplicate]

In the realm, where the sum of natural numbers is $-1/12$ : $1+2+3+4+...=-1/12$ Is this true?: $2+4+6+8+...=2*(1+2+3+4+...)=-2/12$ Can this kind of naive calculations always be done? -or are there ...
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1answer
26 views

Determining convergence/divergence

I am studying for my exam tomorrow and have come across some problems I cannot get. I have put them below with what I have tried/thinking process behind. Thank you. ...
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1answer
38 views

A question to clarify the use of divergent series in calculating the casimir effect

I asked this question already on both Physics SE and quora, but I did not get an answer on either of these Q&A venues. I know this is strictly speaking not a mathematics question, but could the ...
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1answer
51 views

Does a series always diverge if its sequence isn't a null-sequence?

I have the following series: $$\sum_{n = 1}^\infty 2^{n^2}z^n$$ The task is to give its radius of convergence. I solved that one using the root-test and came to the same answer. But the solution ...
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4answers
71 views

Does the following series converge or diverge?

(2n-1)/(n!) I used the ratio test here and got the lim as n -> infinity to be 0. Therefore, I assumed that the series converges. However, my textbook says that it ...
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2answers
57 views

Can divergent series be useful?

As explained by Terence Tao on his blog for example, it is possible to give a value to some divergent series using analytic continuation. For instance, that allows identities like $$\sum\limits_{n ...
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2answers
185 views

Is $1 + 2 + 3 + \dots = -\frac{1}{12}$ really true? [duplicate]

I've read this strange result of the sum of all positive integers being $-\frac{1}{12}$. Is it really true? Does this also mean this is true? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
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0answers
23 views

Infinite series: existence of power to ensure finiteness

Let $(a_n)_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence converging to $0$. Does there always exist a real number $r\in[0,\infty)$ such that $\sum_{n\in\mathbb{N}}|a_n|^r<\infty$? Does such real ...
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0answers
51 views

Showing how this infinite sum diverges [duplicate]

$\displaystyle{\sum_{n=1}^{\infty} \left[\left( 1 + {1 \over n}\right)^{n} - {\rm e}\right]}$ I tried both root and ratio tests (for the root test, the expression became way too complex to handle) ...
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2answers
266 views

Problem in Divergent series

Lets try to evaluate $$\frac{(-1)}{1^s}+\frac{(-1)^2}{2^s}+\frac{(-1)^3}{3^s}+...$$ $$=\frac{e^{\pi i}}{1^s}+\frac{e^{2\pi i}}{2^s}+\frac{e^{3\pi i}}{3^s}+...$$ $$=\frac{1}{1^s}(1+\frac{\pi ...
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0answers
46 views

Dirichlet series summation method

Is it true that if $\lim\limits_{x\to1^-}a_1x+a_2x^2+a_3x^3+...=C$ exists then it is necessarily true that $\lim\limits_{s\to0}\frac{a_1}{1^s}+\frac{a_2}{2^s}+\frac{a_3}{3^s}+... =C$ It seems like ...
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2answers
49 views

Does the Leibniz Test say that if the sequence doesn't go to 0 that the series diverges?

My book explains the Leibniz test by saying: "Assume a sub n is a positive sequence that converges to 0..." And goes on to say that that means the alternating series converges. What if the sequence ...
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1answer
83 views

Is this a valid way to prove that this infinite sum is divergent?

I have the infinite sum: $$\sum_{n=1}^{\infty}\frac{1}{2(n+2)}$$ In this sum I observe that all instances of n is added with 2 before used. Therefor I would think i could do this ...
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0answers
133 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
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27 views

Prove convergence of a serie little with Direct comparison test

I have the following Serie $$\sum_{k=1}^{\infty} \log(1+\frac{1}{k^2})$$ this serie should converge but when i apply the Direct comparison test it diverges $$|\sum_{k=1}^{\infty} ...
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2answers
42 views

Is my divergence test correct?

This idea came to me while looking at the following graph of $f=\frac{1}{x}$: Now, the definite integral of $f$ from $1$ to $n$ is smaller than $f(1)+f(2)...f(n)$, from the graph above. But since ...
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2answers
88 views

If the set of natural numbers is closed under addition, how can we have the result that the sum of all the natural numbers to infinity is -1/12 [duplicate]

As seen here and on this wikipedia page the sum of all the natural numbers to infinity is -1/12. $\sum_{n=1}^\infty n = \frac{-1}{12}$ but the set of natural numbers is closed under addition and ...
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1answer
50 views

Convergence, Divergence and Summability of this series

If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? $\sum_{n=0}^{\infty} ...
2
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3answers
53 views

Series convergence problem

I´m having trouble analyzing the convergence of another series: $$\sum_{n=1}^\infty (-1)^n \frac 1 {n^{\frac 1 n}}$$ If you could just give me a hint about which test for convergence should I use, ...
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3answers
40 views

Convergence of an infinite series problem

I am having trouble with the series $$\sum_{n=1}^\infty (-1)^n\frac n {n+1}$$ I want to know if it converges or not, and I´ve tried with the comparision test, the ratio test, the Leibniz test... ...
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2answers
34 views

Does not converge nor does it diverge to infinity or negative infinity

I am stuck on part D of my problem. Suppose that $a_n$ converges to $0$ and $b_n$ converges to infinity. $c_n = (a_n) \times (b_n)$, Give an example where $(c_n)$ does not converge, nor does it ...
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4answers
70 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
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0answers
104 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
7
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2answers
120 views

Eulers Doubly Infinite Geometric Series

Recently I came across and "Identity" discovered by Euler. $$ E = \cdots + X^3 + X^2 + X + 1 + \frac{1}{X} + \frac{1}{X^2} + \frac{1}{X^3} + \cdots = 0 $$ You can formally obtain this expression by ...