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July 5th, 2011, 03:27 PM   #1
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Series with inverse binomial factor!

[color=#000000]Compute the series .[/color]
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July 7th, 2011, 03:30 AM   #2
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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.
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July 7th, 2011, 04:20 AM   #3
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Re: Series with inverse binomial factor!

[color=#000000] nice!! Now try .[/color]
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July 7th, 2011, 04:45 AM   #4
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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.
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July 7th, 2011, 04:50 AM   #5
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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]
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July 7th, 2011, 05:35 AM   #6
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Re: Series with inverse binomial factor!

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July 8th, 2011, 03:59 PM   #7
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Re: Series with inverse binomial factor!

Quote:
Originally Posted by galactus
How about ?.
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July 9th, 2011, 03:37 AM   #8
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Re: Series with inverse binomial factor!

Quote:
Originally Posted by Stupid_man
Quote:
Originally Posted by galactus
How about ?.

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.
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September 6th, 2011, 04:31 PM   #9
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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]
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September 17th, 2011, 12:28 PM   #10
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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.
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