
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 5th, 2011, 03:27 PM  #1 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Series with inverse binomial factor! [color=#000000]Compute the series .[/color] 
July 7th, 2011, 03:30 AM  #2 
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Series with inverse binomial factor! Use of the Gamma function: Now, use the Beta/Gamma relation: Using the sum of a geometric series, differentiating, then integrating: What is interesting to notice is that the Golden Ratio appears in the evaluation. 
July 7th, 2011, 04:20 AM  #3 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Series with inverse binomial factor! [color=#000000] nice!! Now try .[/color] 
July 7th, 2011, 04:45 AM  #4 
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Series with inverse binomial factor!
In that event, the same previous idea can be used except the integral becomes: How about ?. Or, maybe even change the power of the n to a 2 or whatever. 
July 7th, 2011, 04:50 AM  #5 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Series with inverse binomial factor! [color=#000000]I have the solution, for too, but for the 3rd power or higher no, you give some good ideas...[/color] 
July 7th, 2011, 05:35 AM  #6 
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Series with inverse binomial factor!
deleted

July 8th, 2011, 03:59 PM  #7  
Newbie Joined: Jul 2011 Posts: 17 Thanks: 0  Re: Series with inverse binomial factor! Quote:
 
July 9th, 2011, 03:37 AM  #8  
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Series with inverse binomial factor! Quote:
It involves the use of the polylogarithm. By using the Beta function: So, By using the polylog: , we get: Integration by parts can be used, along with a little knowledge of the polylog, to arrive at the sum. It's rather tedious. If you have another way, I am always open to seeing other methods. The best book on the topic is "Polylogarithms and associated functions" by Leonard Lewin.  
September 6th, 2011, 04:31 PM  #9 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Series with inverse binomial factor! [color=#000000]Galactus I remember we have forgotten something after a long time.......took me months to finally solve it! Computation of . and so hence for x=1 we get, .[/color] 
September 17th, 2011, 12:28 PM  #10 
Senior Member Joined: May 2011 Posts: 501 Thanks: 6  Re: Series with inverse binomial factor!
Very nice, Z. That's a clever approach. Good ol' Beta and Gamma sure come in handy, huh?. Sorry I have not been around for a while. Been very busy and have not been on the sites much lately. If you're interested, and I think you are, here is a general form for a series: If m=2, then we have If m=3, then we have and so on. The integration is not too elementary, though. 

Tags 
binomial, factor, inverse, series 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Boltzmann factor integers? Divide is not inverse of multiply  BenFRayfield  Elementary Math  0  January 18th, 2014 06:37 AM 
binomial series precision  povilasb  PreCalculus  8  December 25th, 2010 10:49 AM 
Use a Binomial series only to calculate the Maclaurin series  maliniarz  Probability and Statistics  1  December 8th, 2010 06:14 PM 
Sum of different primal factor of a series.  amityak  Number Theory  6  November 27th, 2008 05:59 AM 
Sum of different primal factor of a series.  amityak  Applied Math  0  December 31st, 1969 04:00 PM 