Determine whether the following series is convergent or not
$$2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n+1}\sum_{k=1}^{n}\frac{1}{k}$$
My trial : Since I was stuck in this problem, I drew the sequence in computer so that I was convinced it converges to zero. So, In order to use alternating series test, I tried to show $\frac{\sum_{k=1}^{n}\frac{1}{k}}{n+1}$ is decreasing and converge to zero. I can identify the series is decreasing, but I was stuck in showing it converges to zero. Could you give me a few hint to proceed the next step..?
