- Thread starter
- #1

#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

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

- Thread starter Alexmahone
- Start date

- Thread starter
- #1

- Jan 26, 2012

- 268

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

- Jan 26, 2012

- 890

The Abel sum is:Find the Abel sum of 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...

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

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