How would you expand this exactly? Would you use the binomial theorem?

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- Jan 17th 2009, 10:19 AMHarisSeries Expansion

How would you expand this exactly? Would you use the binomial theorem? - Jan 17th 2009, 10:30 AMJhevon
the binomial expansion doesn't give rise to series, it simply shows you how to expand brackets.

you can write the Taylor series for this. perhaps that's what you want? - Jan 17th 2009, 10:40 AMHaris
Well I thought it was either that or the MacLaurin series. Could someone clarify which one it is and maybe give me a head start?

- Jan 17th 2009, 10:51 AMJhevon
A MacLaurin series

*is*a Taylor series. Specifically one centered around zero. It's usually the one people think about when they think Taylor series. The same link i gave you before talks about them. You can also see here.

So begin by finding a few derivatives of the function, and evaluate them at zero. Then plug them into the form you see. - Jan 17th 2009, 11:08 AMHallsofIvy
On the contrary, the

**extended**binomial expansion, for non-integer x, does lead to an infinite series.

The binomial expansions says that

If n is a positive integer then

is defined as and is 0 for i> n so the sum terminates.

If n is not a positive integer, then n! is not defined but we can still right out the last of those- but the sum no longer terminates.

In particular, . - Jan 17th 2009, 11:14 AMHaris

I got this using the MacLaurin series but somehow it doesnt look as though its correct? - Jan 17th 2009, 11:16 AMo_O
Perhaps the OP is referring to the representation of a binomial as an infinite sum: Binomial Theorem

for any and if . Here, let and .

So we have:

See where you can go from here. Try to simplify the general term and hopefully you'll be able to get somewhere (Yes)