Question

Determine if the series is absolutely convergent, conditionally convergent, or divergent. \[\sum_{2}^{\infty}\left[ \left( -2n \right)\div \left( n +1 \right) \right]^{5n}\]

Answer 1

rewrite the function as [(-2n)/(n+1)]^5^n, then apply the root test. The n-th power should disappear, leaving you a tidy limit to solve for

Answer 2

it diverges, by the way
http://www.wolframalpha.com/input/?i=sum+from+2+to+infinity+%28-2n%2F%28n%2B1%29%29^%285n%29

Answer 3

\[\lim_{n\to\infty}\left|\frac{-2n}{n+1}\right|=2>1\]

therefore there is no chance for the original series to converge. Hence it diverges.