I can't make out the summand in (d), and I addressed (c) in your other question.
(a) [tex]\displaystyle\sum_{n\ge1}\frac{\cos n\pi}{n\sqrt n}[/tex]
We have for positive integers [tex]n[/tex] that [tex]\cos n\pi=(-1)^n[/tex]. We also are aware that the series
[tex]\displaystyle\sum_{n\ge1}\frac1{n^{3/2}}[/tex]
converges, since it is a [tex]p[/tex]-series with [tex]p=\dfrac32>1[/tex]. Since the [tex]p[/tex]-series converges in absolute value, the alternating series must also converge by comparison.
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(b) [tex]\displaystyle\sum_{n\ge1}(-1)^n\frac{\ln n}n[/tex]
By the alternating series test, this series will converge if the absolute value of the summand is increasing for some large enough [tex]N[/tex] and approaches zero.
We have
[tex]\left|(-1)^n\dfrac{\ln n}n\right|=\dfrac{\ln n}n>0[/tex]
for all [tex]n\ge1[/tex], and we also have that
[tex]\displaystyle\lim_{n\to\infty}\frac{\ln n}n=\lim_{m\to\infty}\frac m{e^m}=0[/tex]
(where we substituted [tex]m=\ln n[/tex], so that [tex]e^m=n[/tex]).
Therefore (b) also converges.
