tag:blogger.com,1999:blog-6912603287930240451.post6462938788470774958..comments2014-12-22T12:00:15.804ZComments on Noncommutative geometry: Infinitesimal variablesArupnoreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6912603287930240451.post-4657013079661021632007-06-28T01:35:00.000Z2007-06-28T01:35:00.000ZDear Theo,Thanks for mentioning Hardy's nice book...Dear Theo,<BR/>Thanks for mentioning Hardy's nice book on divergent series. In between I read a bit more on Euler's method of computing zeta values.I can identify at least two methods that he used. His early success, as mentioned in Ayoub's article, was as follows, in modern notation. Let <BR/>Li_2 (x)= \sum x^n/n^2<BR/> be the `dilogarithm' function'. He proved the identity<BR/>Li_2 (x)+Li_2masoudnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-51848074625910400882007-06-27T17:13:00.000Z2007-06-27T17:13:00.000ZMasoud,I don't know which procedure Euler used for...Masoud,<BR/><BR/>I don't know which procedure Euler used for zeta(2); he certainly had quite a collection of methods to make slowly-convergent series speed up.<BR/><BR/>The very fine book <I>Divergent Series</I> by G.H. Hardy (1949) discusses a few.Theohttp://www.blogger.com/profile/03344294173628793721noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-35379011704008300762007-06-23T17:28:00.000Z2007-06-23T17:28:00.000ZThere is an interesting entry level Wikipedia arti...There is an interesting entry level Wikipedia article on Euler-MacLaurin summation formula at<BR/>http://en.wikipedia.org/wiki/Euler-Maclaurin_formula<BR/>It points out to the double-edged nature of the formula: to use it to approximate a series by a definite integral, as Euler did, or approximating an integral by a series, as in MacLaurin's case. <BR/><BR/> Nice post!Chrisnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-75234528329673769732007-06-22T19:33:00.000Z2007-06-22T19:33:00.000ZDavid Goss has just kindly pointed out a paper by ...David Goss has just kindly pointed out a paper by Raymond Ayoub (`Euler<BR/>and the Zeta function' American Math Monthly, Dec. 1974) where the issue of numerical computation of zeta values by Euler is explained very well. His early success was in 1731 where he showed zeta (2)= 1.644934 by an elaborate summation technique which increased the rate of convergence. His second computation was in masoudnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-74393212848109809822007-06-22T17:16:00.000Z2007-06-22T17:16:00.000ZHi Alain,Thanks for this post. I just wanted to co...Hi Alain,<BR/>Thanks for this post. I just wanted to comment on Euler's book `Introductio in Analysis Infinitorum'. Indeed the numerical computation of zeta values is another achievement of Euler. The original series for the zeta function is slowly convergent (to find zeta (2) using this series within just six decimal digits one has to add a million terms!) So the original series, it seems to masoudnoreply@blogger.com