Is there a name for infinite series of this type?

$(1 - \frac12)+(\frac13 - \frac14)(1 - \frac12 + \frac13)+(\frac15 - \frac16)(1 - \frac12 + \frac13 - \frac14 + \frac15)+(\frac17 - \frac18)(1 - \frac12 + \frac13 - \frac14 + \frac15 - \frac16 + \frac17)+...$

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?

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Some keywords for such kind of sums of sums [of sums] : Euler or Euler-Zagier sums, Multiple Zeta Value with many papers with the name Broadhurst Borwein and so on, see too Multiple Polylogarithms (in the last one you'll see that alternate sums are considered too). –  Raymond Manzoni Dec 28 '12 at 21:37

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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Ah so it is a double sum. Thought so. Sorry to bother you with such a trivial question it was just that sometimes it's written with the sigma in the middle which somehow confused me! –  KingChem Dec 29 '12 at 0:57