009C Sample Midterm 3, Problem 4

Test the series for convergence or divergence.

(a) (6 points)      ${\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\sin {\frac {\pi }{n}}.$

(b) (6 points)      ${\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\cos {\frac {\pi }{n}}.$

ExpandFoundations:
For $n\geq 2$, both sine and cosine of ${\frac {\pi }{n}}$ are strictly nonnegative. Thus, these series are alternating, and we can apply the
Alternating Series Test: If a series $\sum _{k=1}^{\infty }a_{k}$ is
Alternating in sign, and
$\lim _{k\rightarrow 0}|a_{k}|=0,$
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 ${\displaystyle \lim _{k\rightarrow \infty }a_{k}\neq 0,}$ then the series/sum $\sum _{k=0}^{\infty }a_{k}$ 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:

Expand(a):
Here, we have
$placehold$

Expand(b):

ExpandFinal Answer:

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