Question

Determine whether the series is convergent or divergent \[\sum_{1}^{\infty}\left( n +4^{n} \right)\div \left( n +6^{n} \right)\]

Answer 1

\[\sum_{n=1}^{\infty}\frac{n +4^{n} }{n +6^{n} }\]

\[\leq\sum_{n=1}^{\infty}\frac{4^{n}+4^{n} }{6^{n} }\]

\[\leq2\sum_{n=1}^{\infty}\frac{4^{n} }{6^{n} }\]

\[=2\sum_{n=1}^{\infty}\left(\frac{4}{6}\right)^{n} \] which is a convergent geometric series

Therefore by the comparison test the original test converges

Answer 2

* therefore the original SUM converges