Determine whether the series is absolutely convergent.
the series from 1 to infinity of
10^n/((n+1)4^(2n+1))
There is no alternating term here, so obviously if this fails to converge it does not converge absolutely.
Note $\displaystyle \sum_{n=1}^{\infty}\frac{10^n}{(n+1)4^{2n+1}}=\sum _{n=1}^{\infty}\frac{10^n}{4(n+1)16^n}\leq\sum_{n= 1}^{\infty}\left(\frac{10}{16}\right)^n=\frac{8}{3 }$