Conditional Convergent Power Series and radius of convergence

Question

I'm just not sure about the fifth case which states that CC at L and CC at R, I've failed to find an example for that, I cannot prove it is wrong either, can someone give me some hints?

Answer 1

Consider the power series $f(x)=\sum\limits_{n=0}^{\infty} \frac{(-1)^{\lfloor n/2 \rfloor}}{n+1} z^n$.

The radius of convergence of $f$ is $1$. Then $f(1)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{\lfloor n/2\rfloor}}{n+1}$, which is not AC, but converges: group the terms by consecutive pairs and use the alternating series test : $f(1) = \sum\limits_{n=0}^{\infty} (-1)^n \cdot \big(\frac{1}{n+1}+\frac{1}{n+2}\big)$.

Similarly, $f(-1)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n+\lfloor n/2 \rfloor}}{n+1} = \sum\limits_{n=0}^{\infty} (-1)^n \big(\frac{1}{n+1}-\frac{1}{n+2}\big)=\sum\limits_{n=0}^{\infty} \frac{(-1)^n}{(n+1)(n+2)}$ : the alternating series test also apply.