$\text { Determine whether the series } \sum _ { n = 0 } ^ { \infty } \frac { \cos ( n \pi ) } { n + 3 } \text { converges conditionally or }$ absolutely, or diverges.
A)The series converges absolutely.
B)The series diverges.
C)The series converges absolutely but does not converge conditionally.
D)The series converges conditionally but does not converge absolutely.

Question

Quizplus · Accepted Answer

Note that $\cos(n\pi) = (-1)^n$, so we have $$\sum_{n=0}^\infty \frac{\cos(n\pi)}{n+3} = \sum_{n=0}^\infty \frac{(-1)^n}{n+3}.$$ This series converges conditionally by the Alternating Series Test, as $\frac{1}{n+3}$ is decreasing and approaches 0 as $n	o \infty$. However, if we take the absolute value, we get $$\sum_{n=0}^\infty \left|\frac{(-1)^n}{n+3}ight| = \sum_{n=0}^\infty \frac{1}{n+3},$$ which diverges by the Divergence Test or the Integral Test. Therefore, the series converges conditionally but not absolutely.