# How do you use the direct Comparison test on the infinite series sum_(n=1)^oo(1+sin(n))/(5^n) ?

The value of sin(x) is always between -1 and 1; $\sin \left(n\right) \ge - 1$ means our series is all non-negative terms. And since we have $\sin \left(n\right) \le 1$ for all n, comparing the nth term we get:
$\frac{1 + \sin \left(n\right)}{{5}^{n}} < \frac{2}{{5}^{n}}$
This means our sum is less than ${\sum}_{n = 1}^{\infty} \frac{2}{{5}^{n}}$, which converges as a geometric series with r = 1/5 < 1.