tag:blogger.com,1999:blog-5376628436133716219.post8648102703963314210..comments2024-03-28T14:45:46.850-07:00Comments on Why I hate physics: The Casimir Effect: Ramanujan RevisitedMarty Greenhttp://www.blogger.com/profile/17624084719249673373noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5376628436133716219.post-53850511942425823822014-01-21T02:21:27.402-08:002014-01-21T02:21:27.402-08:00I just use "Ramanujan Series" as a descr...I just use "Ramanujan Series" as a descriptive term for all those growing integer series that have non-intuitive sums, and it's only the alternating ones that I have a method for. As far as the Gaussian envelope, I explain it more in my next blogpost...Marty Greenhttps://www.blogger.com/profile/17624084719249673373noreply@blogger.comtag:blogger.com,1999:blog-5376628436133716219.post-5730221390114386702014-01-21T00:42:12.403-08:002014-01-21T00:42:12.403-08:00Interesting! How does the Gaussian envelope form, ...Interesting! How does the Gaussian envelope form, I mean how do you find the sequence to which you multiply the terms? Can you explain?<br /><br />This seems research worthy!<br /><br />Note : Not all alternating series are by Ramanujan! Some are Abel, some are Grandi, etc.Balarkanoreply@blogger.comtag:blogger.com,1999:blog-5376628436133716219.post-17018573595460720942014-01-20T10:09:32.568-08:002014-01-20T10:09:32.568-08:00Well...one example of a Gaussian envelope would be...Well...one example of a Gaussian envelope would be:<br /><br />0.999 <br />0.996 <br />0.991<br />0.984<br />etc.<br /><br />to a first-order approximation. So multiply these numbers by the terms of the series and you get:<br /><br /> 0.999<br />-1.992<br />+2.973<br />-3.936<br />+4.875<br />etc.<br /><br />I did this in Excel and the running total grows to around +/-7 after 20 terms, and then comes back down, stabilizing at 0.25 after around 120 terms.<br /><br />This Gaussian has a standard deviation of around 20. I can try a tighter Gaussian, with a standard deviation of only 5, and it stabilizes much more quickly (after only 25 terms) but not quite as accurately...to around 0.252...<br /><br />It's funny how well this stuff works. I think it gives you the right answer for any of Ramanujan's alternating series.<br /><br /><br />Marty Greenhttps://www.blogger.com/profile/17624084719249673373noreply@blogger.comtag:blogger.com,1999:blog-5376628436133716219.post-38691723573846747542014-01-20T09:18:19.299-08:002014-01-20T09:18:19.299-08:00Can you explain that Gaussian envelope thing you m...Can you explain that Gaussian envelope thing you mentioned for evaluating 1 - 2 + 3 - 4 + ... ?Balarkanoreply@blogger.com