Question

Determine whether the following series converge, converge absolutely, converge conditionally, or diverge.

1. $\displaystyle\sum_{k=1}^ ∞$ $(√k+1/√k)$^-3
2. $\displaystyle\sum_{k=1}^ ∞$ $(cos(k\pi/2)/(2k)\pi$
3. $\displaystyle\sum_{k=1}^ ∞$(-1)^k+1 $(k-1)!(k+1)!/(2k)!$
4. $\displaystyle\sum_{k=1}^ ∞$ $(1.3.5...(2k-1))/(1.4.4.7...(3k-2)))$

Answer 1

1.

$\int^{\infin}_1\frac{1}{x^p}dx$ is convergent if and only if $p>1$

$\displaystyle{\lim_{k\to \infin}}(\sqrt{k}+1/\sqrt k)^{-3}(\sqrt k)^3=1\neq 0$

Since series $\sum(1/k^{3/2})$ converge, series $\displaystyle\sum_{k=1}^ ∞$ $(√k+1/√k)$^-3 converge also.

2.

$\displaystyle\sum_{k=1}^ ∞$ $(cos(k\pi/2)/(2k)\pi$=$\displaystyle\sum_{k=1}^ ∞$ $\frac{(-1)^{k}\pi}{4k}$

Since $\displaystyle\sum_{k=1}^ ∞$ $\frac{1}{k}$ diverge and $\displaystyle{\lim_{k\to \infin}}|(cos(k\pi/2)/(2k)\pi|=0$ , the series conditionally convergent.

3.

$\displaystyle{\lim_{k\to \infin}}\frac{(k+2)!k!}{(2k+2)!}\cdot\frac{(2k)!}{(k+1)!(k-1)!}=\displaystyle{\lim_{k\to \infin}}\frac{(k+2)k}{(2k+1)(2k+2)}=1/4<1$

Since $\displaystyle\sum_{k=1}^ ∞|a_k|$ converge, the series absolutely converge.

4.

$\displaystyle{\lim_{k\to \infin}}\frac{a_{k+1}}{a_k}=\displaystyle{\lim_{k\to \infin}}\frac{2(k+1)-1}{3(k+1)-2}=2/3<1$

The series converge.