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 Post subject: SeriesPosted: Mon, 9 Jul 2012 04:50:06 UTC
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How to find series representation for quantities such as , , , and ?

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 04:53:49 UTC
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glebovg wrote:
How to find series representation for quantities such as , , , and ?

For use the Cauchy rule for multiplication of power series and pick your favorites for and

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:02:12 UTC
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glebovg wrote:
How to find series representation for quantities such as , , , and ?

For use the Cauchy rule for multiplication of power series and pick your favorites for and

How did you come up with ?

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:16:02 UTC
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glebovg wrote:
glebovg wrote:
How to find series representation for quantities such as , , , and ?

For use the Cauchy rule for multiplication of power series and pick your favorites for and

How did you come up with ?

Just modified the basic by recalling , a simple parlor trick.

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:23:52 UTC
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Also, I do not think the Cauchy product would give me the explicit form for and .

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:33:36 UTC
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glebovg wrote:
Also, I do not think the Cauchy product would give me the explicit form for and .

Very well, here are some other versions:

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:43:13 UTC
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How do you generally come up with series representations, say for ?

I know Ramanujan came up with a lot of formulas for and other quantities without any derivations, but how would you derive series for quantities such as ?

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 05:50:45 UTC
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glebovg wrote:
How do you generally come up with series representations, say for ?

I know Ramanujan came up with a lot of formulas for and other quantities without any derivations, but how would you derive series for quantities such as ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 06:01:02 UTC
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glebovg wrote:
Also, I do not think the Cauchy product would give me the explicit form for and .

Why not?

For example,

are absolutely convergent series, so taking products,

are series representation of and .

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 06:02:02 UTC
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glebovg wrote:
How do you generally come up with series representations, say for ?

I know Ramanujan came up with a lot of formulas for and other quantities without any derivations, but how would you derive series for quantities such as ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

I think Ramanujan was looking at hypergeometric functions or suchlikes.

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 06:04:07 UTC
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Joined: Mon, 28 Dec 2009 00:16:28 UTC
Posts: 228
Thanks for help.

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 Post subject: Re: SeriesPosted: Mon, 9 Jul 2012 06:09:10 UTC
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outermeasure wrote:
glebovg wrote:
How do you generally come up with series representations, say for ?

I know Ramanujan came up with a lot of formulas for and other quantities without any derivations, but how would you derive series for quantities such as ?

Look for functions that have pi as an argument or output that is easily found and get their taylor series.

I think Ramanujan was looking at hypergeometric functions or suchlikes.

Yes, he beat that horse to death a lot.

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