Size of this PNG preview of this SVG file: 512 × 341 pixels. Other resolutions: 320 × 213 pixels | 640 × 426 pixels | 1,024 × 682 pixels | 1,280 × 853 pixels | 2,560 × 1,705 pixels.
Original file (SVG file, nominally 512 × 341 pixels, file size: 54 KB)
 | This is a file from the Wikimedia Commons. Information from its description page there is shown below. Commons is a freely licensed media file repository. You can help. |
Summary
| Description |
Binomial coefficients C (n, k) extended for negative and fractional n, illustrated with expansion of a simple binomial by CMG Lee. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case n = −1 gives Grandi's series. Shaded in blue, n = 0.5 gives : substituting and integrating from 0 to ½ gives half the area of a semicircle minus a circular segment, , yielding Newton's breakthrough method of approximating . |
| Source | Own work |
| Author | Cmglee |
Licensing
I, the copyright holder of this work, hereby publish it under the following licenses:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International 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.
|
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. |
You may select the license of your choice.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 19:04, 29 March 2021 | | 512 × 341 (54 KB) | Cmglee | Remove duplicated elements. |
| 17:59, 28 March 2021 |  | 512 × 341 (295 KB) | Cmglee | Add fractional ''n''. | |
| 03:25, 20 March 2021 |  | 512 × 640 (31 KB) | Cmglee | {{Information |Description=Binomial coefficients ''C''(''n'', ''k'') extended for negative ''n'' illustrated with expansion of a simple binomial by CMG Lee. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case ''n'' = −1 gives Grandi's series. |Source={{own}} |Date= |Author= Cmglee |Permission= |other_versions= }} Category:Binomial coefficients Category:Pascal's triangle |
File usage
The following pages on the English Wikipedia use this file (pages on other projects are not listed):
