Question

use the ratio or root test to determine whether the series is convergent.
sum ((n!)^2)/((2n)!+n^2), n=1 to infinity

Answer 1

\[\sum_{n=1}^{\infty}\frac{ (n!)^2 }{ (2n)!+n^2 }\]

Answer 2

Turns out that first suggestion isn't ideal. Try comparing to \(\dfrac{(n!)^2}{(2n)!}\). Now,
\[\lim_{n\to\infty}\left|\frac{\dfrac{((n+1)!)^2}{(2(n+1))!}}{\dfrac{(n!)^2}{(2n)!}}\right|=\lim_{n\to\infty}\frac{\dfrac{(n+1)^2}{(2n+2)!}}{\dfrac{1}{(2n)!}}=\lim_{n\to\infty}\frac{(n+1)^2}{(2n+2)(2n+1)}\]

Answer 3

thank you