# How do you use the direct Comparison test on the infinite series sum_(n=1)^oo9^n/(3+10^n) ?

Sep 11, 2014

By Comparison Test, we can conclude that the series
${\sum}_{n = 1}^{\infty} \frac{{9}^{n}}{3 + {10}^{n}}$ converges.

Let us look at some details.
For all $n \ge q 1$,
${9}^{n} / \left\{3 + {10}^{n}\right\} \le q {9}^{n} / {10}^{n} = {\left(\frac{9}{10}\right)}^{n}$

By Geometric Series Test,
${\sum}_{n = 1}^{\infty} {\left(\frac{9}{10}\right)}^{n}$ converges since $| r | = \frac{9}{10} < 1$

Hence, by Comparison Test,
${\sum}_{n = 1}^{\infty} \frac{{9}^{n}}{3 + {10}^{n}}$ also converges.