Question

Find the radius of convergence and interval of convergence of the series

\[\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{7n}}{(6n)!}\]

Answer 1

Ive gotten to this step..

\[\lim_{n \rightarrow \infty }\left| x \right|^6 * \frac{(6n)!}{(6n + 6)!}\]

Answer 2

That should be \[\left| x \right|^7 sorry\]

Answer 3

Note that:\[(6n+6)!=(6n+6)(6n+5)(6n+4)(6n+3)(6n+2)(6n+1)(6n)!\]

Answer 4

\[
\frac {(6 n)!}{(6n+6)!}= \frac 1{(6n+1)(6n+2)(6n+3)\cdots (6n+6)}
\]

Answer 5

ok i see

Answer 6

so i would be left with in the denominator??? 6n + 6 ??

Answer 7

You have to take the limit when n goes to \[+ \infty\]