Convergence of alternating series using alternating series test

Question

The series is $\sum\limits_{k=1}^{\infty}\dfrac{(-1)^{2k-1}}{2k-1}$. It satisfies both conditions of convergence: $\lim\limits_{k\to \infty}|u_k|=0$ and $|u_{k+1}|<|u_k|$, so it should converge right?

According to my textbook this diverges. So where did I go wrong?

Answer 1

The criterion you saw in the textbooks is certainly the alternating series criterion. But the series you have is not such a series.

Answer 2

I had to give this multiple looks before I saw the problem.

The problem is that the series does not alternate: $$(-1)^{2k-1}=-1$$ for all $k\in\mathbb{Z}$.

Answer 3

Hints:

1. Since $(-1)^{2k-1}=-1$ for all $k$ we have

$\sum_{k=1}^{\infty} \frac{(-1)^{2k-1}}{2k-1} =\sum_{k=1}^{\infty} \frac{-1}{2k-1} .$

2. $\frac{1}{2k-1} \ge \frac{1}{2} \cdot \frac{1}{k}.$

Can you proceed ?