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.

115 questions
48 views

Improper Lebesgue integral $\int_{\mathbb R}\frac{\sin(x)}x~\mathrm dx$

While thinking over the idea of an improper Lebesgue integral, I came up with the following: $$\int_0^\infty\mu\{x~|~f(x)>t\}-\mu\{x~|~f(x)<-t\}~\mathrm dt$$ with $\mu$ being the Lebesgue ...
30 views

Is square of Poisson kernel conditionally convergent?

Poisson kernel is defined for $\theta\in [0,2\pi)$ as $$P_r(\theta) = \sum\limits_{n\in\mathbb{Z}} r^{|n|}e^{in\theta}$$ which equals $$P_r(\theta) = \frac{1-r^2}{r^2-2r\cos (\theta) + 1 }.$$ For this ...
44 views

Convergence of the series below

$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$ I did: $$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$ $$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$ So diverges by the Ratio ...
32 views

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?
30 views

Could I get an explanation on why this would conditionally converge?

$$\sum\limits_{n=2}^\infty \frac{\cos(n\pi)}{\ln(n)^2}$$ I'm not sure how this would conditionally converge, according to my calculations I would assume it's absolutely converge.
30 views

Numerical Solver Exercise

Sensitivity to initial conditions is well illustrated by a little target practice with your numerical solver. Enter $x' = x^2 - t$ into your numerical solver, and then experiment with initial ...
32 views

In completion to this question: Need A help in understanding a step in matrix representation of bounded linear operators. The book said: "Now, $$A \phi_{j} = \sum_{k}<A \phi_{j},\phi_{k}> \... 2answers 41 views Infinite series convergence question$$\sum_{n=3}^{\infty}\frac{(-1)^n}{\log n}$$Can the conditional convergence of this series be proved by alternating series test, since you need n to be a natural number for the alternating series ... 1answer 207 views Theorem 3.54 Baby Rudin (Riemann's Series Theorem) [duplicate] I am struggling trying to understand the final part of the proof of Theorem 3.54 on Baby Rudin. Here's the Theorem Let \sum a_n be a conditionally convergent series. Suppose :$$ -\infty \leq \...
I'm dealing with the following series $$\sum_{n\ge 2} \frac{1}{2+(-1)^nn}\,.$$ Clearly this will not converge absolutely as the general term in absolute value $\sim 1/n$. Leibniz can not be applied ...