# 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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Let $B_{1},B_{2},\dots,B_{n} \in \mathcal{F}$ partition of $\Omega$ and $\mathcal{P}=\sigma(B_{1},B_{2},\dots,B_{n})$ If $P(B_j)>0 , j = 1,\dots,n$. Then show that : \mathbb{E}[X|\mathcal{P}] = \... 1answer 66 views ### Follow-up question on conditionally convergent series. This is a follow-up from this question. Hypothesis: If \sum\limits_{n=0}^{\infty} a_n\  is a conditionally convergent series with \ \limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1, ... 3answers 69 views ### Is there an ambiguity on \sum\limits^{\infty}_{i=0}a_i? Sorry if this is basic but I saw the theorem that says if \sum\limits^{\infty}_{i=0}a_i is conditionally convergent then this series can be any number up to rearrangement. Then does it imply that \... 1answer 52 views ### Prove specially rearranged alternating harmonic series converges to \frac 12 \ln{\frac{4p}{q}} By Leibnitz's test the alternating series is convergent. \displaystyle \sum_{n=1}^{\infty} (-1)^n \frac 1n =\frac 12 \ln{\frac{4×1}{1}} \begin{align} & \left(1-\frac 12- \frac 14\right)+\left(\... 1answer 61 views ### How can I show that if 2 power series are conditionally convergent then it can happen that \sum_{k=1}^{\infty}c_k diverges? Assume that \sum_{k=1}^{\infty}a_k =A \in R and \sum_{k=1}^{\infty}b_k =B \in R with partial sums r_n =\sum_{k=1}^{n}a_k and s_n=\sum_{k=1}^{n}b_k. If one wants to sum the double series \sum_{... 1answer 46 views ### Absolute convergence of \sum_{n=1}^{\infty} a_nx^n I have to find an a_n such that \sum_{n=1}^{\infty}a_nx^n  converges absolutely on [-1,1]. I have choosen a_n = \frac {1}{n^2} and did the root test and see that \lim_{n \to \infty} (|a_n|) =... 1answer 63 views ### if \sum_{i=1}^n a_i converges but \sum_{i=1}^n (a_i)^2 diverges, does that mean that \sum_{i=1}^n a_i is conditionally [duplicate] I don't think this is true, but cannot find an example to disproof it. Either that or i need to prove that \sum_{i=1}^n |a_i| diverges, which I am unclear how to approach the proof. 2answers 141 views ### Prove that the series \sum\limits_{n=2}^{\infty}(-1)^n\frac{\ln(n)}{n^x} converges to a positive real number for all x > 0 Prove that the series \sum\limits_{n=2}^{\infty} (-1)^n\frac{\ln(n)}{n^x}=\frac{\ln(2)}{2^x} - \frac{\ln(3)}{3^x} + \frac{\ln(4)}{4^x} - \frac{\ln(5)}{5^x} + ... converges to a positive real number ... 0answers 47 views ### Decidability of convergence of real series given an oracle for positive series I've been reading for the last hour a few posts on this site about series that no one knows if they converge or not. I was quite surprised that they were almost all consisting of positive terms. ... 1answer 195 views ### Is the series conditionally convergent, absolutely convergent or divergent \sum(-1)^n\frac{\ln^3 n}n. Determine if the series\sum_{n=1}^\infty(-1)^n\frac{\ln^3 n}n$$is conditionally convergent, absolutely convergent or divergent. By comparison test I got a_n > b_n therefore divergent with ... 3answers 38 views ### Absolute vs Conditional Convergence of an alternating series [closed] Picture of the Alternating Series$$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrtn(\sqrt{n+2})}$$This is the picture of the alternating series I am working with. I found that it is convergent by the ... 1answer 132 views ### The alternating series \sum_{k=1}^{\infty} \frac{(-1)^k(2 - \sin k)}{2k} seems to be convergent, but Leibniz criterion does not apply I was looking for an example of convergent, alternating series \sum_{k=1}^{\infty}(-1)^kb_k such that \{b_k\}_{k=1}^{\infty} is not eventually monotone, so that Leibiniz criterion could not be ... 0answers 98 views ### Laplace Transform of \cos(t)/t this seems like a homework problem. yes! To some extent. But really I was not getting it. I was not able to get the Laplace transform of \cos(t)/t. using the property of Integration in Laplace ... 1answer 314 views ### Convergence of \sum\frac{\sin n\theta}{n^r} and \sum_{n=1}^\infty u_n \cos (n\theta+a). Problem 1. Show qth power of \sum\frac{\sin n\theta}{n^r} (formed by Abel's rule, i.e.$$\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},$$... 1answer 36 views ### Proving convergence in probability functions In the textbook during the steps to expand the convergance of probability the following is provided. the condition for converges in probability is given as \lim_{n\rightarrow \infty}\mathbb{P}(|X_n-X|... 2answers 151 views ### Conditional convergence for improper Riemann double integrals I'm reading Buck's advanced calculus. It says for improper integral of higher dimensions, conditional convergence is impossible, i.e., \int\int_D f cannot exist without \int\int_D|f| existing too. ... 0answers 207 views ### Convergence of Euler product implies convergence of Dirichlet series? (Crossposted to Math Overflow) Suppose we have an Euler product over the primes$$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$where each a_p \in \mathbb{C}. The Euler product is ... 1answer 152 views ### If \sum a_n^k converges for all k \geq 1, does \prod (1 + a_n) converge? By definition, an infinite product \prod (1 + a_n) converges iff the sum \sum \log(1 + a_n) converges, enabling us to use various convergence tests for infinite sums, and the Taylor expansion$$ \...
$\{a_n\}$ is a sequence of real numbers.$\space\sum_{n=1}^{\infty} a_{2n}$ and $\sum_{n=1}^{\infty} a_{2n-1}$ are both conditionally convergent. Is there such $\sum_{n=1}^{\infty} a_{n}$ that is ...
What is the domain of convergence in variable $a$ of the Taylor Series of this function? $$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)ln\left(p\left(x,a\right)\right)dx$$ ...