Proving that a Series Converges Conditionally

Question

So, I was suppose to prove that the series$\sum^{\infty}_{k=1}a_n$ is conditionally convergent given $\{a_n\}$ is the sequence

$$ a_n= \begin{cases} 1/k&\mathrm{if\ }n=2k-1,\\ -1/k&\mathrm{if\ }n=2k.\\ \end{cases} $$

I know that $ a_n=\{1, -1, \frac{1}{2}, -\frac{1}{2},...,\frac{1}{k}, -\frac{1}{k}, \frac{1}{k+1}, -\frac{1}{k+1}, ... \}$ but how do I show that it converges conditionally?

**correction: ** $\{1, -1/2, 1/3, -1/4, ...\}$ Thanks in advance.

Answer 1

Hint:

Conditional convergence means that $\sum^{\infty}_{k=1}a_k$ converges, but $\sum^{\infty}_{k=1}\left|a_k\right|$ doesn't. Obviously,

$$\sum^{\infty}_{k=1}\left|a_k\right| = \sum^{\infty}_{k=1}\frac 1 k$$ is the harmonic series, which is divergent.

To prove the convergence of $\sum^{\infty}_{k=1}a_k$, you can use the alternating series test.