# 009C Sample Midterm 3, Problem 4

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Test the series for convergence or divergence.

(a) (6 points)      ${\displaystyle {\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\sin {\frac {\pi }{n}}.}$
(b) (6 points)      ${\displaystyle {\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\cos {\frac {\pi }{n}}.}$
Foundations:
For ${\displaystyle n\geq 2}$, both sine and cosine of ${\displaystyle {\frac {\pi }{n}}}$ are strictly nonnegative. Thus, these series are alternating, and we can apply the
Alternating Series Test: If a series ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ is
• Alternating in sign, and
• ${\displaystyle \lim _{k\rightarrow 0}|a_{k}|=0,}$
then the series is convergent.
Note that if the series does not converge to zero, we must claim it diverges by the

Divergence Test: If ${\displaystyle {\displaystyle \lim _{k\rightarrow \infty }a_{k}\neq 0,}}$ then the series/sum ${\displaystyle \sum _{k=0}^{\infty }a_{k}}$ diverges.

In the case of an alternating series, such as the two listed for this problem, we can choose to show it does not converge to zero absolutely.

Solution:

(a):
Here, we have
${\displaystyle placehold}$
(b):