Binominal series


by Nikitin
Tags: binominal, series
Nikitin
Nikitin is offline
#1
Dec14-12, 02:05 PM
P: 588
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.
Phys.Org News Partner Mathematics news on Phys.org
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
Pseudo-mathematics and financial charlatanism
Michael Redei
Michael Redei is offline
#2
Dec14-12, 02:25 PM
P: 181
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
jbriggs444 is offline
#3
Dec14-12, 02:26 PM
P: 748
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
Nikitin is offline
#4
Dec14-12, 04:03 PM
P: 588

Binominal series


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
awkward is offline
#5
Dec14-12, 06:19 PM
P: 325
The binomial series for [itex](1+x)^\alpha[/itex] has infinitely many terms except when [itex]\alpha[/itex] is a non-negative integer (even when [itex]\alpha[/itex] is real).


Register to reply

Related Discussions
3 questions from basic binominal theory Precalculus Mathematics Homework 10
Probability - Binominal distribution Precalculus Mathematics Homework 2
binominal theorem Precalculus Mathematics Homework 6
Maximum likelihood estimator of binominal distribution Calculus & Beyond Homework 1
Binominal theorem Precalculus Mathematics Homework 2