|
| If this is your first visit, be sure to check out the FAQ by clicking the link above. You may have to register before you can post: click the register link above to proceed. To start viewing messages, select the forum that you want to visit from the selection below. |
|
|||||||
| Tags: alternating, divergent, series |
|
|
Thread Tools | Display Modes |
|
#1
July 7th 08
posted to sci.math.research
|
|||
|
|||
divergent alternating series
Hello! Ramanujan found the "value" of the divergent alternating series 0^0 - 1^1 + 2^2 - 3^3 + 4^4 - 5^5 +- ... = 0.704169... (see G. N. Watson, Theorems stated by Ramanujan. VIII: Theorems on divergent series, J. London Math. Soc. 4 (1929) 82-86). The constant on the right-hand side is the definite integral from 1 to infinity of the function 1/x^x. In the web page http://mathworld.wolfram.com/Hundred...eProblems.html there is a claim that 2^1 - 4^3 + 6^5 - 8^7 +- ... = 0.323367... Can someone find a reference containing this result, or a correct proof that this is true? Also, is 1^2 - 3^4 + 5^6 - 7^8 +- ... = 0.459360... true as well? Thank you! Steve Finch http://algo.inria.fr/bsolve/ __________________________________________________ _______________ Need to know now? Get instant answers with Windows Live Messenger. http://www.windowslive.com/messenger...enger_ 072008 
|
| Ads |
|
#2
July 7th 08
posted to sci.math.research
|
|||
|
|||
|
divergent alternating series
Am 07.07.2008 16:18 schrieb Steven Finch: Hello! Ramanujan found the "value" of the divergent alternating series 0^0 - 1^1 + 2^2 - 3^3 + 4^4 - 5^5 +- ... = 0.704169... (see G. N. Watson, Theorems stated by Ramanujan. VIII: Theorems on divergent series, J. London Math. Soc. 4 (1929) 82-86). The constant on the right-hand side is the definite integral from 1 to infinity of the function 1/x^x. Some monthes ago I dealt with such series in the context of tetration, and got a value of ~ 0.70368 for this series. This was a first attack at these types of series, and if the above value is correct, then my idea must be flawed in any way. On the other hand, the basic approach is fairly general and pehaps needs only a polishing - I've put it aside last year for later consideration - why not now... I'd put a draft for the "tetration-forum" about this, the style is a bit q&d - maybe you're interested anyway. See http://go.helms-net.de/tetdocs/Tetra_Etaseries.pdf Gottfried Helms
|
|
#3
July 7th 08
posted to sci.math.research
|
|||
|
|||
|
divergent alternating series
Am 07.07.2008 17:24 schrieb Gottfried Helms: is a bit q&d - maybe you're interested anyway. See http://go.helms-net.de/tetdocs/Tetra_Etaseries.pdf Sorry, typo: http://go.helms-net.de/math/tetdocs/Tetra_Etaseries.pdf Gottfried Helms
|
| Thread Tools | |
| Display Modes | |
|
|
Similar Threads
|
||||
| Thread | Thread Starter | Forum | Replies | Last Post |
| [tetration] Alternating series of powertowers of increasing heights/aconjecture | Gottfried Helms | Mathematical Research (Moderated) | 1 | November 23rd 07 12:12 PM |
| Does anybody know how to evaluate this summation of alternating series? | Vista | Physics - General Discussion | 14 | July 7th 07 04:56 PM |
| Nonrenormalization vs Renormalization 25.3: Alternating Series and Trilinearity | OsherD | Physics - General Discussion | 1 | April 28th 06 07:00 PM |
| Probable Influence/Causation Versus Algebraic Geometry/Topology 5: Alternating Series | OsherD | Physics - General Discussion | 1 | February 12th 06 02:37 AM |
| QFT, divergent power series, and all that... | Frank Hellmann | Current Physics Research (Moderated) | 22 | May 29th 04 05:54 PM |
Linear Mode
