$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 10 ^ { n } } \text { using any }$ appropriate test.
A) diverges
B) converges

Question

$\text { Determine the convergence or divergence of the series } \sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 10 ^ { n } } \text { using any }$ appropriate test. 
A) diverges
B) converges

Quizplus · Accepted Answer

One way to approach this problem is to use the comparison test, comparing the given series with a geometric series.

Notice that $|\cos n| \leq 1$ for all $n \geq 1$. Therefore, we have

$$\left|\frac{\cos n}{10^n}\right| \leq \frac{1}{10^n}$$

for all $n \geq 1$.

Now, consider the geometric series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{10^n}$. This is a convergent series with sum $\frac{1}{1-\frac{1}{10}} = \frac{10}{9}$.

Since $\displaystyle \left|\frac{\cos n}{10^n}\right| \leq \frac{1}{10^n}$ for all $n \geq 1$, and $\displaystyle \sum_{n=1}^{\infty} \frac{1}{10^n}$ converges, we can use the comparison test to conclude that $\displaystyle \sum_{n=1}^{\infty} \frac{\cos n}{10^n}$ converges.

Therefore, the answer is B.