Questions tagged [conditional-convergence]

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.

198 questions
28
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4 answers

Why do we ask for *absolute* convergence of a series to define the mean of a discrete random variable?

If $X$ is a discrete random variable that can take the values $x_1, x_2, \dots $ and with probability mass function $f_X$, then we define its mean by the number $$\sum x_i f_X(x_i) $$ (1) when the series above is absolutely convergent. That's the…
22
votes
1 answer

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$?

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? In other words, does there exist an infinite sequence $(a_k)_{k \in \mathbb N_0}$ such that $$e^x = a_0 + \sum_{1 \leq k < \infty} a_k \left(1 + \frac x k\right)^k$$ for all $x…
18
votes
1 answer

Power series which diverges precisely at the roots of unity, converges elsewhere

Is there a complex power series $\sum a_nz^n$ with radius of convergence $1$ which diverges at the roots of unity (e.g., $z=e^{2\pi i\theta}$, $\theta \in \mathbb{Q}$) and converges elsewhere on the unit circle ($z=e^{2\pi i\theta}$, $\theta \in…
16
votes
3 answers

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose $$-\infty \leq \alpha \leq \beta \leq +\infty.$$ …
15
votes
2 answers

When $\Big[ uv \Big]_{x\,:=\,0}^{x\,:=\,1}$ and $\int_{x\,:=\,0}^{x\,:=\,1} v\,du$ are infinite but $\int_{x\,:=\,0}^{x\,:=\,1}u\,dv$ is finite

I have encountered a simple problem in probability where I would not have expected to find conditional convergence lurking about, but there it is. So I wonder: Is any insight about probability to be drawn from the fact that infinity minus infinity…
15
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3 answers

Rearrangements that never change the value of a sum

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_k = \lim_{n\to\infty} \sum_{k=1}^n a_{f(k)}, $$ where "$=$" is construed as meaning that…
12
votes
2 answers

Conditionally convergent power sums

I'm struggling on the following question: Let $S$ be a (possibly infinite) set of odd positive integers. Prove that there exists a real sequence $(x_n)$ such that, for each positive integer $k$, the series $\sum x_n^k$ converges iff $k \in…
11
votes
0 answers

Rearranging series and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. Let $S_\infty$ be the group of permutations of $\mathbb{N}$. For…
11
votes
1 answer

Is it possible that $\sum a_{\sigma(n)}$ converges iff $\sum b_{\sigma(n)}$ diverges for every permutation $\sigma :\mathbb N\to \mathbb N$?

Here is my question: Is it possible to find a pair of series $\sum a_n, \sum b_n,$ each having rearrangements that converge conditionally, such that $\sum a_{\sigma(n)}$ converges iff $\sum b_{\sigma(n)}$ diverges for every permutation $\sigma…
10
votes
3 answers

How do I manipulate the sum of all natural numbers to make it converge to an arbitrary number?

I just found out that the Riemann Series Theorem lets us do the following: $$\sum_{i=1}^\infty{i}=-\frac{1}{12}$$But it also says (at least according to the wikipedia page on the subject) that a conditionally convergent sum can be manipulated to…
8
votes
4 answers

Why does the commutative property of addition not hold for conditionally convergent series?

I learned about the Riemann rearrangement theorem recently and I'm trying to develop an intuition as to why commutativity breaks down for conditionally convergent series. I understand the technique used in the theorem, but it just seems really odd…
7
votes
1 answer

How to use AND condition in Desmos

Sorry maybe it's not typical mathematics question, but Desmos is very helpful in solving and testing mathematics issues, so maybe anyone could help me. I can't figure it out how to use AND condition in Desmos For example to make OR you can just use…
pajczur
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6
votes
2 answers

Does the series $\sum_{n=1}^{\infty}\sin\left(2\pi\sqrt{n^2+\alpha^2\sin n+(-1)^n}\right)$ converge?

Let $\alpha$ be such that $0\leq \alpha \leq 1$. Since $\sin n$ has no limit as $n$ tends to $\infty$, I'm having trouble with finding if the series $$\sum_{n=1}^{\infty}\sin \left(2\pi\sqrt{n^2+\alpha^2\sin n+(-1)^n}\right)$$ is convergent? Thanks.
6
votes
2 answers

Is it possible for a power series to be conditionally convergent at two different points?

Like I stated in the title, I was just wondering if it's possible for a power series to be conditionally convergent at two different points. Are there any examples of power series that fit this criteria? Any help is appreciated!
6
votes
2 answers

For any conditionally convergent series $\sum _{n=1}^\infty a_n,\ \exists\ k\geq 2\ $ such that the subseries $\sum _{n=1}^\infty a_{nk}$ converges.

A subseries of the series $\displaystyle\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\displaystyle\sum _{k=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove or disprove: For any conditionally convergent series…
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