Why does the ln(x) series converge for x=2 but not x=0?

[schaefera]

Homework Statement

This isn't really so much a homework problem as me asking a question. The Taylor Series for ln(x) centered at 1 is: sum_[0, infinity] ((-1)^((n+1)*(x-1)^n))/n, then why does it converge for the endpoint x=2, but not x=0 of the interval of convergence?

Homework Equations

The Attempt at a Solution

Letting x=2, you get the alternating series (-1)^(n+1)/(n), which converges by the alternating series test.
Letting x=0, don't you get (-1)^(2n+1)/n, which should also converge as an alternating series?

 

[Dick]

(-1)^(2n+1) does not alternate. It always has the same sign.

 

[schaefera]

And now I just feel silly! I'm so used to a series having a (..+1) in the exponent to alternate, looks like I forgot how to multiply n by 2! This is what happens when I have to worry about infinity.

Thanks!