Thread: 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

Since the limit is zero it means the interval of convergence is the entire number line.

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.

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.

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

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.