# Interval of convergence (calculus)

• Apr 24th 2007, 04:29 PM
faure72
Interval of convergence (calculus)
The following problem was puzzling me so any help would be greatly appreciated!

Find the interval of convergence for the following series:

sigma starting at j=0 and going to infinity of:
(-1)^j times (x)^(2j) divided by j factorial
• Apr 24th 2007, 04:37 PM
ThePerfectHacker
Since the limit is zero it means the interval of convergence is the entire number line.
• Apr 24th 2007, 04:41 PM
Jhevon
Quote:

Originally Posted by ThePerfectHacker
Since the limit is zero it means the interval of convergence is the entire number line.

really? i don't recall that rule.
• Apr 24th 2007, 04:42 PM
ThePerfectHacker
Quote:

Originally Posted by Jhevon
really? i don't recall that rule.

We need to find all the x such that the limit is strictly less than 1.

We do that by the ratio test. But since the limit is always zero, not matter what x, it means it is strictly less than 1. Thus, it converges for all x.
• Apr 24th 2007, 04:46 PM
Jhevon
Quote:

Originally Posted by ThePerfectHacker
We need to find all the x such that the limit is strictly less than 1.

We do that by the ratio test. But since the limit is always zero, not matter what x, it means it is strictly less than 1. Thus, it converges for all x.

yeah, i get you

thanks
• Apr 24th 2007, 06:06 PM
clockingly
Convergence problem
Using the ratio test, you wind up with lim of x^2 divided by k+1 as j approaches infinity. This limit will be 0 regardless of what x equals.