Question

Test the series for convergence or divergence.

Answer 1

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

Answer 2

Does not converges .....

Answer 3

i know but how can i prove it?

Answer 4

do you know what
\[\frac{k!}{(k+2)!}\] is ?

Answer 5

no

Answer 6

write out the first five terms of each and it will be obvious

Answer 7

what do you mean?

Answer 8

write out the first five terms of \((k+2) !\) and the first 3 terms of \(k!\) and see what they are
then it will be clear

Answer 9

You could also try the ratio test:
Consider \[ a_k=\frac{2^k(k!)}{(k+2)!}\] verify that
\[\lim_{n\rightarrow\infty}\left| \frac{a_{k+1}}{a_k}\right|>1\], which means that the sum will diverge.

Answer 10

that should say \(\lim_{k\rightarrow\infty}\) rather than \(n\)