Question

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

Answer 1

By Comparison Test, we can conclude that the series
`sum_{n=1}^infty{9^n}/{3+10^n}` converges.

Let us look at some details.
For all `n geq 1`,
`9^n/{3+10^n} leq 9^n/10^n=(9/10)^n`

By Geometric Series Test,
`sum_{n=1}^infty(9/10)^n` converges since `|r|=9/10<1`

Hence, by Comparison Test,
`sum_{n=1}^infty{9^n}/{3+10^n}` also converges.