I asked a question about this series;
$(1 - \frac12)+(\frac13 - \frac14)(1 - \frac12 + \frac13)+(\frac15 - \frac16)(1 - \frac12 + \frac13 - \frac14 + \frac15)+(\frac17 - \frac18)(1 - \frac12 + \frac13 - \frac14 + \frac15 - \frac16 + \frac17)+...$
in a previous thread and something else about it that I'd like to know is if there is a name for series where the coefficient of each term is a partial sum? Furthermore, is there a general method for finding the closed form sums of such series?
Answer
There is no special name since what you have is just a double summation instead of a single summation. Your series is nothing but
$$\sum_{n=0}^{\infty} \sum_{k=1}^{2n+1} \left(\dfrac1{2n+1}-\dfrac1{2n+2}\right) \left(\dfrac{(-1)^{k-1}}{k} \right)$$
which can also be written as a single summation
$$\sum_{n=0}^{\infty} \left(\dfrac1{2n+1}-\dfrac1{2n+2}\right) \left(H_{2n+1} - H_n\right)$$
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