apply the ratio test and find if it converges: the sum from n=1 to infinity of (n^90)/n!

Answer 1

Ratio test states that we compute lim_n->∞|a_n+1/a_n| where a_n=n⁹⁰/n! and a_n+1=(n+1)⁹⁰/(n+1)!

lim_n->∞|a_n+1/a_n|

= lim_n->∞|[(n+1)⁹⁰/(n+1)!]/[n⁹⁰/n!]|

= lim_n->∞|[(n+1)⁹⁰/((n+1)n!)]/[n⁹⁰/n!]|

= lim_n->∞|(n+1)⁸⁹/n⁹⁰|

= 0<1

Since the limit is less than 1, this means that ∑[n=1,∞] n⁹⁰/n! absolutely converges, which also implies that it conditionally convergences as well (although we don't necessarily say that in this case).

Hope this helped!

Answer 2

See the video.