# 009C Sample Midterm 3, Problem 4 Detailed Solution

Test the series for convergence or divergence.

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

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

Background Information:
Alternating Series Test
Let  ${\displaystyle \{a_{n}\}}$  be a positive, decreasing sequence where  ${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0.}$
Then,  ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}}$  and  ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}}$
converge.

Solution:

(a)

Step 1:
First, we note that
${\displaystyle \sin {\bigg (}{\frac {\pi }{n}}{\bigg )}>0}$
for all  ${\displaystyle n\geq 1.}$
So, the series
${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}\sin {\bigg (}{\frac {\pi }{n}}{\bigg )}}$
is alternating.
Let  ${\displaystyle b_{n}=\sin {\bigg (}{\frac {\pi }{n}}{\bigg )}.}$
Step 2:
The sequence  ${\displaystyle \{b_{n}\}}$  is decreasing since
${\displaystyle \sin {\bigg (}{\frac {\pi }{n+1}}{\bigg )}<\sin {\bigg (}{\frac {\pi }{n}}{\bigg )}}$
for all  ${\displaystyle n\geq 2.}$
Also,

${\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }b_{n}}&=&\displaystyle {\lim _{n\rightarrow \infty }\sin {\bigg (}{\frac {\pi }{n}}{\bigg )}}\\&&\\&=&\displaystyle {\sin(0)}\\&&\\&=&\displaystyle {0.}\end{array}}}$

Therefore,
${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}\sin {\frac {\pi }{n}}}$
converges by the Alternating Series Test.

(b)

Step 1:
First, we note that
${\displaystyle \cos {\bigg (}{\frac {\pi }{n}}{\bigg )}>0}$
for all  ${\displaystyle n\geq 3.}$
So, the series
${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}\cos {\bigg (}{\frac {\pi }{n}}{\bigg )}}$
is alternating.
Also, we have

${\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }\cos {\bigg (}{\frac {\pi }{n}}{\bigg )}}&=&\displaystyle {\cos(0)}\\&&\\&=&\displaystyle {1.}\end{array}}}$

Step 2:
Since  ${\displaystyle \lim _{n\rightarrow \infty }\cos {\bigg (}{\frac {\pi }{n}}{\bigg )}\neq 0,}$  we have
${\displaystyle \lim _{n\rightarrow \infty }(-1)^{n}\cos {\bigg (}{\frac {\pi }{n}}{\bigg )}=DNE.}$
Therefore, the series diverges by the Divergence Test.