Why does the binomial series have an infinite number of terms?

[Nikitin]

Hello. I'm revising the material in preparation for the exam, and I found something I fail at understanding.

When defining binomial series, http://en.wikipedia.org/wiki/Binomial_series, why is the sum of the binomial "(m k)" going from 1 to ∞? Shouldn't it instead be going from 1 to m (the function can only be differentiated m times)?

Afterall, binomial series are a form of taylor series, and a taylor series of a function can't have infinite terms when the function can only be differentiated a finite amount of times.

 

[Michael Redei]

Which function can only be differentiated m times? The binomial series is the Taylor series at x=0 of (1+x)^α for some complex number α. And (1+x)^α can be differentiated infinitely often, unless α is a non-negative integer.

 

[jbriggs444]

Just because all but finitely many terms are zero does not mean that you cannot consider a series as having infinitely many terms. The function f(x) = (1+x)^m can be differentiated more than m times. It's just that all of the derivitives are eventually zero.

It looks to me like a choice to make the summation look more like the generic form of the Taylor series (which it is, after all) rather than an equally accurate truncation thereof.

 

[Nikitin]

Michael: oops, sorry. My book used the notation of m instead of a, and I have little knowledge of complex numbers (I'm only doing my 1st semester).

jbriggs: yeh, so it's just a formal thing? Allright, that's good enough 4 me :)

 

[awkward]

The binomial series for $(1+x)^\alpha$ has infinitely many terms except when $\alpha$ is a non-negative integer (even when $\alpha$ is real).