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Identity

Member

Registered: 2007-04-18

Posts: 934

Pi Identity

Hi, this could be a hard question:

In the 18th century, Euler proved the remarkable fact that:

Use this to determine the value of:

Thanks, if you can't help directly then please give me some additional info from links.

Last edited by Identity (2007-04-26 19:40:41)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Pi Identity

Since the series is absolutely convergent, we can rearrange the order of summation in the series.

Last edited by JaneFairfax (2007-04-26 21:17:12)

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Pi Identity

Lovely!

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Pi Identity

The problem is very easy if you know how Euler solved the Basel problem (finding the exact value of 1/1² + 1/2² + 1/3² + ...). He cleverly manipulated the Taylor expansion of sin x. To solve your problem, you do virtually the same thing but with the Taylor expansion for cos x.

We start with the Maclaurin series for cos x:

This function will have zeros at (2n+ 1)π, so factor it according to that fact:

This simplifies to

Now if you were to multiply this all out, the coefficient of x² would be

But in our original expression for cos x, the coefficient for x² was 1/2! = 1/2, so the sum must be equal to 1/2:

Now, simplify:

So

Ninja edit: My solution is more creative than Jane's.

Real edit: But Jane's uses the fact that 1/1² + 1/2² + 1/3² + ... = π²/6, so that might be better to use if this is for an assignment.

Last edited by Zhylliolom (2007-04-26 21:29:45)

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Pi Identity

Jane:
Elegant solution, the part where you made the reciprocated even squares 1/4 of the reciprocated integer squares was a surprise. Did you just look at it and say, "Ah, this must be 1/4 of that?", or did you have to think hard about it... I would never have guessed... thanks anyhow.

Zhylliolom:
Very nice... getting technical! I understood all of it except the root finding factorisation and the original series.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Pi Identity

Zhylliolom's factorization is very interesting- from infinite entries to infinite factors!

I know it's from the rule for 1-3x+2x[sup]2[/sup]= (1-2x)(1-x) sort of thing.

But still I am surprised to find that this rule applies to a polynomial of Infinite degree as well!!!

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X'(y-Xβ)=0