Show in each case that the series converges or diverges

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

(a)

We have for positive integers that . We also are aware that the series



converges, since it is a -series with . Since the -series converges in absolute value, the alternating series must also converge by comparison.

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(b)

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

We have



for all , and we also have that



(where we substituted , so that ).

Therefore (b) also converges.