Abel Summation

Calculus Level 2

12+34+56+1-2+3-4+5-6+\cdots

Define the Abel sum n=0an\sum\limits_{n=0}^\infty a_n to be limz1n=0anzn, \displaystyle \lim_{z\to 1^-} \sum_{n=0}^\infty a_nz^n, if that limit exists.

The Abel sum of the (divergent) series as shown above can be written as ab \frac{a}{b}, where a a and b b are coprime positive integers. Find a+b a+b.

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