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.
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Find all the values $x\in\mathbb{R}$ where the Series converge
$$
\sum_{n=1}^{\infty}\frac{\sqrt{n}+3n}{2^{n}+5n}(x-1)^{n}
$$
I calculate the limit $n\to\infty$ with the D'Alambert ratio test, and the series converges in the interval $-1<x<3$:
$$
\sum_{n=1}^...
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Proof verification: Show that if a series is conditionally convergent, then the series from its positive terms is divergent.
Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.
Please, help me to ...
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Absolutely vs Non-Absolutely Convergent Infinite Product
Consider the following standard infinite product:
$$
\prod_{n=1}^{+\infty} \left( 1 + \frac{(-1)^n}{2n-1}z \right)
$$
This product is not absolutely convergent because:
$$
\sum_{n=1}^{+\infty} \left|...
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Convergence of Riemann zeta function [duplicate]
I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
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Conditional convergent series implies existence of rearrangement that diverges: Doesn't the sum of the negative terms tend to $-\infty$?
In the proof for the Riemann Series Theorem that I'm reading, the author is currently establishing the existence of a divergent rearrangement of an infinite series given that the original series ...
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how do you compute the value of $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$
I know that the series $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging.
I try to evaluate out ...
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Rearranging conditionally convergent series without changing the limit
Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that the series $\sum a_n$ is conditionally convergent, i.e. the limit $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
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To decompose a conditionally convergent series into a partial bounded series and another decreasing series
In the first-year mathematics analysis course, the instructor assigned a problem on the convergence of series. We are given that a series $\sum_{n=1}^\infty A_n$ converges absolutely if $\sum_{n=1}^\...
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Does $a_n$ converge if $a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$? [closed]
Does $a_n$ converge if$a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$?
As described in the title, it seems intuitively that it should converge, but I don't know how to ...
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Find the conditional expectation $E[X \mid X \leq p]$
Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X \...
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Convergence of Alternating Series Involving Cosine Term and Square Root
I am working on a series and attempting to determine its conditional convergence using the alternating series test.
$$
\sum_{n=1}^{\infty} \frac{\cos\left(\frac{\pi}{4} + 2\pi n\right)}{\sqrt{n}}
$$
I ...
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1
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For conditionally convergent series $\sum a_n,\exists (m_n)_{n\in\mathbb{N}}$ with $(n-1)k<m_n\leq n k,$ s.t. $\sum_{n\in\mathbb{N}} a_{m_n}=\alpha.$
Properties of conditionally convergent series $\ \displaystyle\sum a_n\ $:
The sum of the positive terms is $+\infty;\ $ the sum of the negative terms is $-\infty.$
$\displaystyle\lim_{\substack{ { n\...
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Re-ordering conditionally convergent series over $\Bbb C$.
For a conditionally convergent series $\sum a_n$ over $\Bbb C$, let
$M\subset \Bbb N$ be a set such that the sub-series
$\displaystyle \sum_{n\in M} a_n$ converges absolutely.
$\sigma:\Bbb N \to \...
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What happens to EX if E|X| is infinity?
---------original question----------------
According to my professor, we can divide $X$ into $X_+=\max(X,0)$ and $X_-=-\min(X,0)$, both nonnegative.
And we have $EX=EX_+-EX_-$ and $E|X|=EX_++EX_-$
For ...
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Can a real function have convergence that oscillates depending on its derivative?
I recently read a post on here in which a user asked if there existed a function,
$$\lim_{x\rightarrow\infty}f(x)$$ is convergent, but;
$$\lim_{x\rightarrow\infty}f'(x)$$
does not converge.
I wanted ...
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