Tagged Questions

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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For bounded sequences, does convergence of the Abel means imply that for the Cesàro means?

See the title. This is true if the sequence is nonnegative; some Tauberian theorems which I was able to find give some more general sufficient conditions. I would like to know if this is true for ...
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Does the family of series have a limit?

For $r<1$ define $F(r)=\sum_{n\in\mathbb N}(-1)^nr^{2^n}$. Does $F$ have a limit as $r\nearrow 1$?
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On the limit of a product of two sequences

If $b_n$ is decreasing and positive and $\sum b_n$ converges and $1/a_n$ is decreasing and positive and $\sum 1/a_n$ diverges, must $\lim \limits_{n \to \infty} a_n b_n=0$?
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Dense subseries of divergent series

Suppose $\sum_{n>1} a_n=\infty$ and $0<a_{n+1}<a_n$. Let $b_k=a_k$ or $b_k=0$ for all integers $k$. Let $R=\lim_{n\rightarrow\infty}((1/n)\sum_{q=1}^{q=n} b_q/a_q)$ If $R>0$, how to show ...
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Trying to revert a series with problematic log term

I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert, $y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$ I know the inverse exists ...
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Divergence of subsequences

How can we show that if $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f(n+1) \geq f(n)>1$ for all $n\ge 1$ and $\sum_{n = 1}^\infty \frac{1}{f(n)} = \infty$, then for all integers $k>1$ we have ...
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Slowing down the divergence

Is there any unbounded function $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f(n+1) \geq f(n)>1$, such that for every $g:\mathbb{N} \rightarrow \mathbb{N}$ with $g(n+1) \geq g(n)>1$ and ...
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A question about the interval of convergence for alternating series

Say we are given the simple power series $$\sum_{i=0}^{\infty}(-1)^k\frac{(x-4)^k}{2^k}$$ The interval of convergence can easily be shown to be $x\in(2,6)$ using the Root Test, and, since absolute ...
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Many convergent sequences imply the initial sequence zero?

In connection to this question, I found a similar problem in another Miklos Schweitzer contest: Problem 8./2007 For $A=\{a_i\}_{i=0}^\infty$ a sequence of real numbers, denote by ...
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Reverse ErdÅ‘s-Turan

If $\{a_i\}$ is a set of positive integers which contains arbitrary long arithmetic progressions, how to show that $\sum\limits_{i=1}^{\infty}\frac{1}{a_i}=\infty$?
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Are there complex Bernoulli numbers?

I am aware of the generalized Bernoulli numbers, but these are not what I'm looking for. I was wondering if there exists such a thing as fractional, real or even complex Bernoulli numbers ( $B_z$ for ...
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Why does $1+2+3+\dots = {-1\over 12}$?

$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$ $\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ ...
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Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
when we say $1+r+r^2+r^3+\cdots=S$ do we mean $S$ is the partial sum of first $n$ terms of this geometric series where $n$ goes to infinity (which is not boundless due to the existence of $n$th term) ...