Question

1 + 2!x + (4!x^2)/(2!)^2 + (6!x^3)/(3!)^2 + (8!x^4)/(4!)^2 + (10!x^5)/(5!)^2

what is the radius of convergence?

Answer 1

\[1+2!x+\frac{4!x^2}{(2!)^2}+\cdots=\sum_{n=0}^\infty\frac{(2n)!x^n}{(n!)^2}\]

Answer 2

Use the ratio test:
\[\lim_{n\to\infty}\left|\frac{(2(n+1))!x^{n+1}}{((n+1)!)^2}\times\frac{(n!)^2}{(2n)!x^n}\right|=|x|\lim_{n\to\infty}\frac{(2n+2)(2n+1)}{(n+1)^2}\]
What are the convergence conditions for the ratio test?

Answer 3

If you've ever used the ratio test, you should know this. If the limit is less than 1, the series is said to converge. If it's greater than 1, it diverges.

What do you get for
\[\lim_{n\to\infty}\frac{(2n+2)(2n+1)}{(n+1)^2}~~?\]

Answer 4

okay well the radius of convergence is 1/4 so I figured it out thanks

Answer 5

You're welcome