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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### Need a second help in understanding a step in matrix representation of bounded linear operators.

In completion to this question:
Need A help in understanding a step in matrix representation of bounded linear operators.
The book said:
"Now, $$A \phi_{j} = \sum_{k} \phi_{k}......(2),$$
Combining (1) (where (1) is $Ax =…

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### Rearrangements and unconditional convergence

Let us consider the convergente series: $∑_{n=1}^{∞}a_{n}=S$ where $S$ is a real number. I know about Rearrangements and unconditional convergence (https://en.wikipedia.org/wiki/Absolute_convergence). But I have the following problem: Is it possible…

Safwane

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### Absolute and conditional convergence of integral

I have the following improper integral:
$$
\int \limits_{1}^{\infty}\left(\cos\left(\frac{\sin(x)}{\sqrt{x}}\right) - \cos\left(\frac{\cos(x)}{\sqrt{x}}\right)\right)dx
$$
And I need to figure out, whether this integral converges absolutely,…

puhsu

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### How to determine Absolute Convergence and Conditional Convergence.

In this hw problem it is asking to determine the absolute convergence values and place them in interval notation same as for conditional convergence.
$$\sum_{n=1}^{\infty}\frac{x^n}{n}$$
My question is where would I go from this? (Hints would be…

Carlos V

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### Prove $\sum_{n = 0}^{\infty} a_n X^n$ converges at every point in $[-1,1)$ if $a_n$ is non-increasing and converges to $0$

Prove that if $a_n$ is non-increasing, converges to $0$, and radius of convergence of $\sum_{n = 0}^{\infty} a_n X^n$ is equal to $1$ then this series converges at every point in $[-1,1)$
I am trying to work out this proof using abel's theorem, but…

e2DAeyePi

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### Conditional and absolute convergence of the integral depending on the parameter

How to determine values of $p$ making the integral
$$\int_{x=2}^{\infty}\frac{(x+1)^{p}\sin(x)}{\log(x)}\mathrm{d}x$$
converges absolutely or conditionally? Comparison $\log(x)$ with $x$ where $x\to\infty$ gives divergence for $1-p<1$. Am I right…

Alex Z.

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### On absolute and conditional convergence of trigonometric integral

I am trying to make out if $\int_1^{\infty}{\arctan(\frac{\cos(x)}{x^{2/3}})}dx$ converges absolutely or conditionally. My answer is that there is no convergence at all, because substitution $x^{\frac{2}{3}}=u$ gives…

Alex Z.

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### Determine if $\sum_{t=1}^\infty (-1)^{n+1}\frac{(-4)^n}{n4^n}$ converges or diverges.

Determine if $$\sum_{t=1}^\infty (-1)^{n+1}\frac{(-4)^n}{n4^n}$$ converges or diverges.
To make it simpler to deal with, I managed to simplify the sum to $$\sum_{t=1}^\infty (-1)^{2n+1}\frac{1}{n}$$ but I can't seem to find a way to show if it…

BB8

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### Sierpinski's generalization of Riemann's rearrangement theorem

I recall that Riemann's theorem on conditionnaly convergent series states that if a series converges conditionnaly, then you can rearrange the sequence so that the new series converges to whatever real number you choose, or diverges.
I'm interested…

Augustin

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### Rearrangement of series and consequences of Riemann's Theorem.

I've been struggling with the following exercise. We're doing the consequences of Riemann's theorem for series, and I've proved that a rearrangement of a serie can diverge to $-\infty$ and $+\infty$, but I don't know how to deal with this…

Relure

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### Cauchy product of an absolutely convergent series and a conditionally convergent series

We know by Mertens' theorem, Let $(a_n)_{n≥0}$ and $(b_n)_{n≥0}$ be real or complex sequences, then if $\sum_{n=0}^{+\infty}a_n$ converges absolutely and $\sum_{n=0}^{+\infty}b_n$ converges only conditionally, then their Cauchy product…

cxh007

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### The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0?

The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0 ?
How do i show for which x the series is conditionally convergent?
Do i have to express in terms of a and b. or something like that?

Danielvanheuven

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### Conditional convergence of a series involving $sin n \theta$

I recently stumbled upon the series $$\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin n \theta}{n}.$$ Consider all values of $\theta$ except $k \pi$ where $k$ is an integer.
Is the series conditionally convergent?
I know…

user321656

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### Convergence of series $\sum_{n =1}^{\infty}\sin(\frac{\pi\cdot n}{4})\cdot \sqrt[9]{\ln(\frac{n+12}{n+9})}$

$$\sum_{n =1}^{\infty}\sin\left(\frac{\pi\cdot n}{4}\right)\cdot \sqrt[9]{\ln\left(\frac{n+12}{n+9}\right)}$$
How to find convergence of this series?
I researched the absolute convergence and get $$\exp^{\frac{1}{3\cdot (n+9)}}$$
Thanks a lot!

user448072

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### Hi there. I have a problem with uniform convergence of functional series.

I have no idea what to do and how to start even, please give some pieces of advice. And if you can solve this task I will be very thankful
$$\sum_{n=1}^{\infty}(\frac{\sin(nx)}{\sqrt[]{n^{3}+1}},\quad x\in (-\infty; \infty)
$$

Ilya Telefus

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