Power series with interval of convergence of $(-1,1]$?

Question

Is there a power series with an interval of convergence of $(-1,1]$? Wouldn't the fact that absolute convergence implies regular convergence make such a function impossible to find?

Answer 1

Take a function that is well-behaved in $1$, but has a singularity in $-1$, then you have a good chance.

$$\log (1+x) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}x^k$$

works. For $x = 1$, you get the alternating harmonic series, which converges (conditionally), and for $x = -1$ you get the negative of the harmonic series which diverges.