MHB Find Abel Sum: Add 1 +1 -1 -1 +1 +1

[alexmahone]

Find the Abel sum of 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...

 

[CaptainBlack]

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

 

[Greg Bernhardt]

Wow, this is a tricky one! The Abel sum of this series is actually undefined because it does not converge to a specific value. This is because the series alternates between adding and subtracting 1, which means it will never settle on a final sum. Some people argue that the Abel sum is 1/2, but others argue that it is actually 1. It's a hotly debated topic in the math community! What do you think the Abel sum should be?