Question

the sum from m=1 to infinity of ((4m-2)/(4m-1)) x^m find the largest interval of convergence for the power series
I know the answer is -1<=x<=1 but how do you get that

Answer 1

Use the ratio test to find that the radius of convergence is 1.
Notice the series does not converges at 1 and -1.
So the interval of convergence is -1 < x < 1

Answer 2

but doesn't the ratio test show that it converges to x

Answer 3

It shows that it converges for |x| < 1, i.e., the ratio test has to be less than 1. So, we our end points, but we still have to test for the endpoints of the interval (-1 and 1, in our case). That's what @eliassaab mentioned in the answer :-)