This tag is for questions related to conditional convergence. A series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

# Questions tagged [conditional-convergence]

198 questions

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### If $\sum a_n$ is convergent, is $\sum\frac{a_n}{n}$ absolutely convergent?

Assume that a series $\sum\limits_{n\geqslant1} a_n$ is convergent. Does this imply that $\sum\limits_{n\geqslant1}\frac{a_n}{n}$ is absolutely convergent?
My first thought is no, but I'm having a hard time finding a counterexample. If there is a…

Alex Mathers

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### Convergence of iterative map

I have the following iterative mapping: $$x_{n+1} = (N-n)^{-1} \frac{x_n}{f(x_n)} \left(C - \sum_{i=1}^n f(x_i)\right)$$ defined for $n \leq N$ and where $C > 0$ is some constant. I am trying to analyze the potential convergence to fixed points or…

Doc Stories

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### Can (the partial sums of) a conditionally convergent series always be written as an alternating sequence of decreasing terms?

True or false:
If $\ \sum a_n\ $ is conditionally convergent series, then there exists
an increasing sequence of integers $\ k_1,\ k_2,\ k_3,\ldots\ $ such
that
$$\ \left(\ \displaystyle\sum_{n=1}^{k_1} a_n\ ,\ \sum_{n=k_1+1}^{k_2}
a_n\ ,\…

Adam Rubinson

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### Laplace Transform of $\cos(t)/t$

this seems like a homework problem. yes! To some extent. But really I was not getting it.
I was not able to get the Laplace transform of $\cos(t)/t$.
using the property of Integration in Laplace Transform,
but since the integral is not convergent,…

crabNebula

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votes

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### What is the domain of convergence of this Taylor Series?

What is the domain of convergence in variable $a$ of the Taylor Series of this function?
$$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)ln\left(p\left(x,a\right)\right)dx$$ …

Jerry Guern

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### Does $\iint_D \frac{x^2}{x^2+y^2} dx dy $ converge on $D= \left\{ (x, y) : x^2+y^2\leq ax \right\} $ ? If yes, what value does it converge to?

Powers equal to $2$ entice the polar substitution.
$D= \left\{ (x, y) : x^2+y^2\leq ax \right\}$, so $0 \leq r \leq a \cos \phi.$ For the domain to make sense, we need either $\phi \in [0, \frac{\pi}{2}] \cup [\frac{3 \pi}{2}, 2\pi)$ and $a \geq 0$,…

fragileradius

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### critical case in $L^p$ convergence

Let $f_n$ be a bounded sequence in $L^1\cap L^{p}$ with $1

Muniain

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### Is there an intuitive way of thinking why a rearranged conditionally convergent series yields different results?

We know that any conditionally convergent series can be made to converge to anything, or even diverge. My question is if we can intuitevely explain why such a thing happens. One would be tempted to think that "eventually, we're summing the same…

Francisco José Letterio

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### If a series diverges, what happens when we divide by partial sums squared?

I'm trying to prove or disprove:
It is given:
$$a_n>0, \sum_{n=0}^\infty a_n=\infty$$
Prove / disprove:
$$\sum_{n=0}^\infty \dfrac{a_n}{S_n^2} <\infty$$
where
$$S_n=\sum_{k=1}^n a_k$$
I tried to prove using all the convergence tests I know, and…

Antonio AN

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votes

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### Conditional convergence of Fourier transform

Suppose $f\in L^1(\mathbf{R})$. Recall the Fourier transform of $f$ is given by
$$
\hat f(x) = \int_{-\infty}^\infty f(t) e^{-2\pi i x t}\, \mathrm{d} t .
$$
The Fourier transform extends from $L^1\cap L^2$ to a unitary operator $L^2 \to L^2$. I am…

Daniel Miller

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### Complex sequence $(z_i)$ such that $\sum_i z_i^k$ converges and $\sum_i \vert z_i \vert^k$ diverges for all $k$

How to find a complex sequence $(z_i)$ such that
$$\sum_i z_i^k \text{ converges for all } k \in \mathbb N$$ but
$$\sum_i \vert z_i \vert^k \text{ diverges for all } k \in \mathbb N ?$$

mathcounterexamples.net

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### Series with digammas

(Inspired by a comment in answer https://math.stackexchange.com/a/699264/442.)
corrected
Let $\Psi(x) = \Gamma'(x)/\Gamma(x)$ be the digamma function. Show
$$
\sum_{n=1}^\infty…

GEdgar

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votes

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### conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$

prove that the series
$$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$
is conditionally convergent?
I tried to prove that it is not absolutely convergent series by trying to prove that $\sum_{n=2}^{\infty} \frac{\vert\cos(n)\vert}{n}$ is divergent, but I…

Jasmine

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votes

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### $1 + {1 \over 3} - {1 \over 2} + {1 \over 5} + {1 \over 7} - {1 \over 4} + {1 \over 9} + {1 \over 11} - {1 \over 6} + +-...$ conditionally convergent

$$1 + {1 \over 3} - {1 \over 2} + {1 \over 5} + {1 \over 7} - {1 \over 4} + {1 \over 9} + {1 \over 11} - {1 \over 6} + +-...$$
I want to show first that $S_{3n}$, $S_{3n+1}$ and $S_{3n+2}$ converges to the same limit, I show
$$S_{3n} = (1 + {1…

Stabilo

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### A problem in convergence and limit

For any non-negative integer $n$ and some finite $r$, we introduce the notation $n_k$ which indicates the number of $\{X_1, X_2, \cdots , X_n\}$ belonging to the $k$-th distribution type, for $k=1, 2, \cdots, r$. Here, it is assumed that $X_i$'s are…

Statistics

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