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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An example of infinite divergent series giving rational fraction of Pi.
Can an example of divergent integer sequence along some regularization method be found where the generalized sum is $c π^k $, k integer and c being a rational?
I only know π appearing, but not as ...
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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 ...
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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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Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$
I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$.
In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
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Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
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the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $
I was looking at the zero's of
$$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$
for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$.
Now this clearly relates to the Riemann zeta:
$$f(s,a) + \...
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Does the limit of the exponential mobius exponential series asymptotically equal its regularized power series?
Context:
Consider the function $\sum_{n=0}^{\infty} e^{nx}$. An extremely unrigorous manipulation of this series would yield
$$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
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Show that $\sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}$ with $ \epsilon > 0$ diverges [duplicate]
I think this sum diverges but I can't seem to show it.
$$ \sum_{n \ge 1} (n \log n)^{-(1-\epsilon)}, \ \ \ \epsilon > 0 $$
I have tried using the bound $n \log n = \log n^n < n^n$ which led me ...
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"Boundary" between convergent and divergent series of the form 1/n^m.
Since $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, but $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, is there some $m \in \mathbb{R}$, with $1 < m < 2$ that defines the "boundary" ...
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Cesaro $(C,\alpha)$ summable implies Abel summable.
I've found quite a few questions regarding the statement "Cesaro $(C,1)$ summability implies Abel summability", e.g. this question, but haven't been able to find a proof for higher Cesaro ...
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Show that for $(u_{n})$ strictly decreasing sequence tending to 0, $\sum_{n=0}^{+\infty} \frac{u_{n}-u_{n+1}}{u_{n+1}}$ diverges
I come to ask you about a problem coming from a serie's exercice sheet brought by a student that I can't crack.
Let $(u_{n})$ be a strictly positive decreasing sequence which converges to 0.
How do I ...
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How can we discuss the "divergent-ness" of an infinite series?
There are many infinite series that converge "regularly" to a finite value, such as the geometric series
$$ \sum^\infty_{n=0}\frac{1}{2^n} = 1+\frac{1}{2} + \frac{1}{4} + \cdots = 2. $$
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