Question

Show in each case that the series converges or diverges

Answer 1

I can't make out the summand in (d), and I addressed (c) in your other question.

(a) $\displaystyle\sum_{n\ge1}\frac{\cos n\pi}{n\sqrt n}$

We have for positive integers $n$ that $\cos n\pi=(-1)^n$. We also are aware that the series

$\displaystyle\sum_{n\ge1}\frac1{n^{3/2}}$

converges, since it is a $p$-series with $p=\dfrac32>1$. Since the $p$-series converges in absolute value, the alternating series must also converge by comparison.

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(b) $\displaystyle\sum_{n\ge1}(-1)^n\frac{\ln n}n$

By the alternating series test, this series will converge if the absolute value of the summand is increasing for some large enough $N$ and approaches zero.

We have

$\left|(-1)^n\dfrac{\ln n}n\right|=\dfrac{\ln n}n>0$

for all $n\ge1$, and we also have that

$\displaystyle\lim_{n\to\infty}\frac{\ln n}n=\lim_{m\to\infty}\frac m{e^m}=0$

(where we substituted $m=\ln n$, so that $e^m=n$).

Therefore (b) also converges.