Math Is Fun Forum

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Closed form for generating functions

How do I show the following:

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Closed form for generating functions

It comes readily from the series expansion of log(1+x) and log(1-x)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Closed form for generating functions

Hi;

Run these.

Assuming[x != 1, 
 FullSimplify[
  Log[Sqrt[(1 + x)/(
    1 - x)]] == (1/2) Log[(1 + x)/(1 - x)] == (1/
     2) (Log[1/(1 - x)] + Log[1 + x])]]

a = Series[Log[1/(1 - x)], {x, 0, 10}] // Normal
b = Series[Log[1 + x], {x, 0, 10}] // Normal

(1/2) (a + b) // Expand

Now match that up term by term with your series on the LHS.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Closed form for generating functions

Last edited by thickhead (2016-12-25 19:14:40)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Closed form for generating functions

Thank you guys

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.