File:Pm1234 Abel.svg

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Original file (SVG file, nominally 197 × 477 pixels, file size: 943 KB)

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Summary

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Description
English: 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 + · · ·.
Date
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Author User:Melchoir
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Licensing

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I, the copyright holder 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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attribution share alike 		This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
This licensing tag was added to this file as part of the GFDL licensing update.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic, 2.0 Generic and 1.0 Generic license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
You may select the license of your choice.

File history

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Date/TimeThumbnailDimensionsUserComment
current02:50, 6 March 2007Thumbnail for version as of 02:50, 6 March 2007197 × 477 (943 KB)Melchoir (talk | contribs)tweak, shorten
01:51, 4 March 2007Thumbnail for version as of 01:51, 4 March 2007145 × 534 (961 KB)Melchoir (talk | contribs)Abel summation of 1 − 2 + 3 − 4 + · · ·

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