TITLE: Calculating $\sum\limits_{n=1}^\infty\frac{{1}}{n+3^n} $
QUESTION [7 upvotes]: I was able to prove this sum  
$$\sum_{n=1}^\infty\frac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.

REPLY [3 votes]: What we can obtain, also by hand, is a reasonable estimation for example by
$$\sum_{n=1}^\infty\frac{{1}}{n+3^n}\approx 0.392<\sum_{n=1}^3\frac{{1}}{n+3^n}+\sum_{n=4}^\infty\frac{{1}}{3^n}=$$$$=\frac14+\frac1{11}+\frac1{30}+\frac32-1-\frac13-\frac1{9}-\frac1{27}\approx 0.393$$