find the radius and interval of convergence for the series

the series from n=1 to infinity of
((-1)^(n+1)*x^n)/n!

I did the ratio test so I had the Lim as n approaches infinity of -x/(n+1), but this is 0, giving no radius, so I think I did something wrong...

To find the radius and interval of convergence for the given series, we need to use the ratio test correctly. Let's go through the steps together.

The ratio test states that for a series ∑(a_n), if the following limit exists:

L = lim(n→∞) |(a_(n+1)) / a_n|

Then the series converges absolutely if L < 1 and diverges if L > 1. If L = 1, the test is inconclusive.

Now, let's apply the ratio test to the given series:

a_n = ((-1)^(n+1) * x^n) / n!

a_(n+1) = ((-1)^(n+2) * x^(n+1)) / (n+1)!

L = lim(n→∞) | a_(n+1) / a_n |
= lim(n→∞) | ((-1)^(n+2) * x^(n+1)) / (n+1)! | / | ((-1)^(n+1) * x^n) / n! |
= lim(n→∞) | (-1) * x / (n+1) |
= |x| * lim(n→∞) 1 / (n+1)
= |x| * 0
= 0

Since the limit of L is zero, the series converges for all values of x. This means that the interval of convergence is (-∞,∞).

As for the radius of convergence, in this case, there is no defined radius since the series converges for all x. In general, the radius of convergence (r) is given by the formula:

r = 1 / lim(n→∞) | a_(n+1) / a_n |

But since the ratio test yielded L = 0, the radius is infinite.

Therefore, the interval of convergence is (-∞,∞) and the series converges for all values of x.