Binomial Series

Key Questions

  • How do you use the binomial theorem to find the Maclaurin series for the function y=f(x)y=f(x) ?

    Binomial Series

    (1+x)^{alpha}=sum_{n=0}^infty((alpha),(n))x^n,

    where ((alpha),(n))={alpha(alpha-1)(alpha-2)cdot cdots cdot(alpha-n+1)}/{n!}.

    Let us look at this example below.

    1/{sqrt{1+x}}

    by rewriting a bit,

    =(1+x)^{-1/2}

    by Binomial Series,

    =sum_{n=0}^infty((-1/2),(n))x^n

    by writing out the binomial coefficients,

    =sum_{n=0}^infty{(-1/2)(-3/2)(-5/2)cdots(-{2n-1}/2)}/{n!}x^n

    by simplifying the coefficients a bit,

    =sum_{n=0}^infty(-1)^n{1cdot3cdot5cdot cdots cdot(2n-1)}/{2^n n!}x^n

    I hope that this was helpful.

    Wataru · 9 · Sep 28 2014
  • What is the link between binomial expansions and Pascal's Triangle?

    Pascal's triangle gives the binomial coefficients.

    Pascal's Triangle

    1
    1 1
    1 2 1
    1 3 3 1
    1 4 6 4 1
    ...

    Binomial Coefficients

    ((0),(0))

    ((1),(0)) ((1),(1))

    ((2),(0)) ((2),(1)) ((2),(2))

    ((3),(0)) ((3),(1)) ((3),(2)) ((3),(3))

    ((4),(0)) ((4),(1)) ((4),(2)) ((4),(3)) ((4),(4))
    ...

    I hope that this was helpful.

    Wataru · · Oct 19 2014
  • What is the formula for binomial expansion?

    To understand Ismail's answer, it is worth recalling some notations:

    ((n),(k))=(n!)/((n-k)!k!), where n,k in NN

    n! =n.(n-1)...2.1

    Reginaldo D. · 1 · Oct 27 2014

Questions

Power Series