# 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.

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### How to prove $\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz$ in the sense of Borel summation?

As the title shows, I would like to prove this identity in the sense of Borel summation, $$\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz,$$ providing ...
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### Does the rate $\log n$ imply "almost harmonic"?

Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$ I have found a reference where it says that the converse is true. I tried to tackle ...
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### Alternating series comparison test

Let's say I have two alternating series of terms, $(-1)^n A_n$ $(-1)^n B_n$ If I know (by for example Leibniz criteria) that one of the series converges / diverges, can I use comparison criteria to ...
1 vote
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### $\sum 2^{-r_n}/r_n$ diverges $\implies$ $\sum 2^{-\lceil r_n \rceil} / {\lceil r_n \rceil}$ diverges

I want to prove $\sum 2^{-r_n}/r_n$ diverges $\implies$ $\sum 2^{-\lceil r_n \rceil} / {\lceil r_n \rceil}$ diverges where $r_n$ is a nondecreasing sequence of reals. This came up in Billingsley ...
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### Elliptic integral singular expansion

The question. Consider the Elliptic Integral $$F(x;k)=\int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}.\tag{1}\label{1}$$ I am interested in the singular series expansion of $F(1;k)$ about $k=1$. I was ...
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$$\sum_{n=2}^{\infty}{\frac{(-i)^{n}}{\ln n}}$$ In the answer, using comparison test $1/\ln n > 1/n,$ the series is divergent. But, in my opinion, the series can be divided like $$\sum_{n=1}^{\... • 21 -2 votes 1 answer 104 views ### How to evaluate \sum \frac{1}{2^{n-r}}\frac{1}{n}? [closed]$$\sum_{n=r+1}^{\infty}\frac{1}{2^{n-r}}\frac{1}{n}$$,where r is a positive integer. First, I wanna know if this is convergent(I guess so) and WHY. And if so, how to evaluate it? • 185 0 votes 0 answers 10 views ### Concoct an alternating series which diverges that is alternating, the limit goes to 0, but is NOT always decreasing. [duplicate] Basically need an example of a divergent series that is alternating and the limit goes to 0. 4 votes 1 answer 85 views ### Divergence of Mehler's Hermite polynomial series According to Mehler's formula,$$ \sum\limits_{n=0}^{\infty}\left(\cfrac{\rho}{2}\right)^n\cfrac{H_n(x)H_n(y)}{n!}=\cfrac{1}{\sqrt{1-\rho^2}}\exp\left(-\cfrac{\rho^2(x^2+y^2)-2\rho x y}{1-\rho^2}\...
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Find the power series representation for 𝑓(𝑥)=1/x+2 cant i write the power series as $$f(x) = \frac{1}{x+2}.$$ so I am a little confused because cant I write the series as  f(x) = \frac{1}{1-(-x-...
If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence positive series $\sum_{n=1}^{\infty} b_n$,which satisfies $\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$ If ...