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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### Does the ordering of a Schauder basis matter in Hilbert space?

If $S=\{v_i\}_{i\in\mathbb N}$ is a (not necessarily orthogonal) Schauder basis for a Hilbert space $H$, must $S$ be an unconditional Schauder basis? I define these terms below because not every source I have found agrees perfectly on the…

WillG

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### Rearrangement in proof for Euler's formula

In the proof for Euler's formula, we expand $e^{ix}$ as a Taylor series, rearrange the terms, factor out $i$, and thus obtain the Taylor series for $\sin (x)$ and $\cos(x)$. However, this rearrangement can only be done if the Taylor series for…

Expain

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### Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples!
Perhaps finding divergent series with absolutely convergent Cauchy product isn't that difficult…

Tony Ma

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### Exercise 9, Chapter 2 of Stein's Fourier Analysis. Showing that a fourier series does not converge absolutely but converges conditionally.

Let $f(x)=\chi_{[a,b]}(x)$ be the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$.
Show that if $a\neq -\pi$, or $b\neq \pi$ and $a\neq b$, then the Fourier series does not converge absolutely for any $x$. [Hint: It suffices to…

nomadicmathematician

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### Can the Cauchy product of a conditionally convergent series with itself be absolutely convergent?

If $\sum_{n\ge 0} a_n$ and $\sum_{n\ge 0} b_n$ are two series, their Cauchy product is defined as $\sum_{n\ge 0} c_n$, where $c_n = \sum^n_{k=0} a_k b_{n-k}$.
As this question points out, finding two conditionally convergent series whose Cauchy…

Jianing Song

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### Convergence of Euler product implies convergence of Dirichlet series?

(Crossposted to Math Overflow) Suppose we have an Euler product over the primes
$$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$
where each $a_p \in \mathbb{C}$. The Euler product is convergent in the range $Re(s) > \sigma_c$, and…

Rivers McForge

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### Sum of series with conditional convergence

Sorry for this question, but for some reason I'm stuck on this for few hours already. Before I solved more complex ( I think ) problems, but can't solve this. The only thing I know that this series converge conditionally, but that is just thing you…

teasu873

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### Is $\sum\limits_{n = 1}^\infty (-1)^n\frac{H_n}{n}=\frac{6\ln^2(2)-\pi^2}{12}$ a valid identity?

A while ago I asked a question about an identity that I found while playing with series involving harmonic numbers. However, since the methods I used were not the focus of my question, I never explained how I got this result. So here it…

Dom

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### Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?

Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?
The ratio test is inconclusive.
It passes the term test for $\Re(n)=1$, but this is not sufficient to prove convergence.
Since we are dealing with so many complex numbers, I do…

Simply Beautiful Art

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### Combining terms in a conditionally convergent series

I am aware that one is unable to rearrange terms in a conditionally convergent series. But, take a conditionally convergent series, say
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$
and group terms with a stride of two to…

s3rgei

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### Convergence of $\sum\frac{\sin n\theta}{n^r}$ and $\sum_{n=1}^\infty u_n \cos (n\theta+a)$.

Problem 1.
Show $q$th power of $\sum\frac{\sin n\theta}{n^r}$ (formed by Abel's rule, i.e. $$\nu_n=\sum_{i_1, i_2,\dots,i_q=n} \frac{\sin i_1\theta}{{i_1}^r}\dots\frac{\sin i_q\theta}{{i_q}^r},$$ where $i_j\in\mathbb{Z}_+, r>0, \theta\in\mathbb{R}$)…

Charlie Chang

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### If $\sum a_n^k$ converges for all $k \geq 1$, does $\prod (1 + a_n)$ converge?

By definition, an infinite product $\prod (1 + a_n)$ converges iff the sum $\sum \log(1 + a_n)$ converges, enabling us to use various convergence tests for infinite sums, and the Taylor expansion
$$
\log(1 + x) = x - x^2/2+x^3/3-x^4/4 +…

Rivers McForge

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### Examples of improper integrals conditionally convergent

I've been looking for examples of improper integrals conditionally convergent, but I get the same result in every document I read:
$$\int_0^\infty \frac{\sin x} x \, dx$$
Can you help me?.

Enrique GF

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### Does the series $\sum^\infty_{n=1} \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ diverge, converge, or converge absolutely?

Problem statement:
Does the series $$\sum^\infty_{n=1} \frac{(-1)^n}{n^{1+\frac{1}{n}}}$$ diverge, converge, or converge absolutely?
EDIT: I appreciate all the answers so far. In my book, we haven't reached the Dirichlet test, Leibniz test, or…

EssentialAnonymity

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### Using Dirichlet's test

I am trying to find out if the following series converges or diverges. If it converges I then want to find out if it converges absolutely or conditionally.
$$\sum _{n=1}^{\infty}\frac{\sin(n)}{\sqrt n}.$$
For the first part I used Dirichlet's law.…

fr14

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