Expert: Paul Klarreich - 8/26/2009

Question
Hi Paul,
What is the Maclaurin series for (1+x)^n also known as binomial series? Can you derive that series?Then give it's particular form in summation notation by letting n=2, n=3 and n=1/2.
Thanks!

Katia

Answer
Questioner: katia
Country: United States
Category: Calculus
Private: No
Subject: Math
Question: Hi Paul,
What is the Maclaurin series for (1+x)^n also known as binomial series? Can you

derive that series?
Then give it's

-- You mean its.  [No apostrophe, please.]

particular form in summation notation by letting n=2, n=3 and n=1/2.
Thanks!

Katia
....................................
When you apply the usual binomial expansion to (1 + x)^n, it looks like this:

SUM(k = 0 to n)[  C(n,k) x^(n - k) ],

but it really is:

SUM(k = 0 to infinity)[  C(n,k) x^(n - k) ],

where C(n,k) is the usual binomial coefficient:
        (n)(n-1)(n-2)(n-3)...(n-k+1)
C(n,k) = ----------------------------
          (k)(k-1)..(3)(2)(1)

When n is a positive integer, nice things happen:

--  each C(n,k) is an integer.
--  the series terminates because once k reaches n+1, the top has (n - n)

somewhere, which is zero.


But if  n is either:
--- a negative number
--- a fraction

it develops into an infinite series, and some of the terms can be fractions.