File:Pm1234 Abel.svg

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Pm1234_Abel.svg‎ (SVG file, nominally 197 × 477 pixels, file size: 943 KB)

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Description	Pm1234 Abel.svg 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 $1-2x+3x^2-4x^3+\cdots.$ 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 + · · ·.
Date	2007-03-04
Source	User created
Author	User:Melchoir
Permission (Reusing this image)	See below.
Other versions	PNG version

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I, the copyright holder of this work, hereby publish it under the following licenses:

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".

This file is licensed under the Creative Commons Attribution ShareAlike 3.0 License. In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it only under a license identical to this one. Official license

This licensing tag was added to this file as part of the GFDL licensing update.

This file is licensed under the Creative Commons Attribution ShareAlike 2.5, Attribution ShareAlike 2.0 and Attribution ShareAlike 1.0 License. In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it only under a license identical to this one.

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	02:50, 6 March 2007		197×477 (943 KB)	Melchoir (Talk \| contribs)	(tweak, shorten)
	01:51, 4 March 2007		145×534 (961 KB)	Melchoir (Talk \| contribs)	(Abel summation of 1 − 2 + 3 − 4 + · · ·)

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File:Pm1234 Abel.png

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Usage of Pm1234 Abel.svg on enwiki
- 1 − 2 + 3 − 4 + · · ·
Usage of Pm1234 Abel.svg on nowiki
- 1 − 2 + 3 − 4 + · · ·
Usage of Pm1234 Abel.svg on ptwiki
- 1 − 2 + 3 − 4 + · · ·

File:Pm1234 Abel.svg

From Wikimedia Commons, the free media repository

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