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

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Answer 1 · 2014-09-11T17:12:10

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.