Question

Help with finding the radius of convergence?

Answer 1

\[\sum_{n=1}^{\infty} 8^n*x^n*n!\]

Answer 2

Using the ratio test,
\[\lim_{n\to\infty}\left|\frac{8^{n+1}x^{n+1}(n+1)!}{8^nx^nn!}\right|=8|x|\lim_{n\to\infty}\left|n+1\right|\]

Answer 3

So the interval of convergence would be 0, making the radius 0?

Answer 4

Uhm, no, not quite. What's the (approaching) value of the limit?

Answer 5

It should be going towards infinity as n goes toward positive infinity

Answer 6

Right, and what are the stipulations of the ratio test?
\[\text{If the limit is _____, then the series converges.}\]

Answer 7

*Otherwise it diverges.

Answer 8

is it really \(\sum_{n=1}^{\infty} 8^n\times x^n\times n!\) ? is there no fraction involved?

Answer 9

Doesn't the limit have to be less than 1 to converge?

Answer 10

or could it be \[\sum_{n=1}^{\infty}\frac{ 8^nx^n}{n!}\]

Answer 11

It would diverge if the limit was greater than 1

Answer 12

Yeah, that's right.