Slika:Pm1234 Abel.svg

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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
    1-2x+3x^2-4x^3+\cdots.
  • 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 + · · ·.

[edit] Licensing

I, the author of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

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