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[edit] Summary
Abel summation of 1 − 2 + 3 − 4 + · · ·.
- The horizontal axis is for a real variable x that takes values between 0 and 1, sometimes including the endpoints. The vertical axis is for various functions of x; it is unbounded, and only the interesting part is shown.
- The black and blue curves are the partial sums — which are functions of x — of the power series

- Functions whose endpoints at x = 1 fall within the diagram are shown in color, and the endpoint is indicated with a dot. The blue dots are the first four partial sums of 1 − 2 + 3 − 4 + · · ·, namely 1, −1, 2, and −2.
- (Only the first twenty or so functions are depicted; if they were all shown, and they all had the same thickness, they would fill the right side of the working area in solid black.)
- For each x < 1, the black and blue curves converge toward the thick purple curve, barely visible in the center of the mess, which depicts the function 1/(1 + x)2.
- The endpoint of the purple curve is a green dot, which falls on the green line at 1/4. This is the Abel sum of 1 − 2 + 3 − 4 + · · ·.
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