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Find the Abel sum

Alexmahone

Active member
Jan 26, 2012
268
Find the Abel sum of 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Find the Abel sum of 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...
The Abel sum is:

\[ A=\lim_{z\to 1}\; \left[1+z-z^2-z^3+z^4+... \right] \]

Now since the series inside the square brackets is absolutly convergent on the interior of the unit disk we may rewrite it as we please:

\[ A=\lim_{z\to 1}\; \left[1+z-z^2-z^3+z^4+... \right]=\lim_{z\to 1}\; \left[(1+z)+(-1)z^2(1+z)+ ... + (-1)^kz^{2k}(1+z)+... \right] \]

so:

\[ A=\lim_{z\to 1}\; \left[(1+z)\sum_{k=1}^{\infty}(-1)^kz^{2k} \right] \]

The sum in the last equation is a convergent geometric series ...

CB