1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is binomial theorem

  1. Jul 23, 2014 #1
    Definition/Summary

    The binomial theorem gives the expansion of a binomial [itex](x+y)^n[/itex] as a summation of terms. The binomial theorem for positive integral values of 'n', is closely related to Pascal's triangle.

    Equations

    The theorem states, for any [itex]n \; \epsilon \; \mathbb{N}[/itex]

    [tex](x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 +......+\binom{n}{n}x^0y^n[/tex]

    In summation form,
    [tex](x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r}y^r[/tex]


    Cases

    1. Substituting y=-y we get,

    [tex](x-y)^n = \binom{n}{0}x^ny^0 - \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 -......+ (-1)^{n}\binom{n}{n}x^0y^n[/tex]

    2. Having y=1 gives,


    [tex](x+1)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1} + \binom{n}{2}x^{n-2} +......+ \binom{n}{n}x^0[/tex]

    Extended explanation

    Proof by Induction

    When [itex]n=0[/itex], the statement obviously holds true, giving [itex](x+y)^0= \binom{0}{0}=1[/itex]

    Assuming it to be true for [itex]n=k[/itex]

    [tex](x+y)^n = \sum_{r=0}^{k} \binom{n}{k} x^{n-k}y^k[/tex]

    Now it needs to hold for [itex]n=k+1[/itex] to complete the inductive step. We use

    [tex](x+y)^{k+1} = x(x+y)^k + y(x+y)^k[/tex]

    Expanding each [itex](x+y)^k[/itex] individually, multiplying by x and y respectively,

    [tex](x+y)^{k+1} = \sum_{r=0}^{k} x^{k+r-1}y^r + \sum_{r=0}^{k} x^{k+r}y^{r+1}[/tex]

    Using the property,

    [tex]\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}[/tex]

    We get,
    [tex](x+y)^{k+1} = \sum_{r=0}^{k+1} \binom{k}{r} x^{(k+1)-r}y^r[/tex]

    This completes our inductive step, proving the theorem.


    Generalization

    For any value of 'n', whether positive, negative, or fractional, the binomial expansion is given by,

    [tex](x+y)^n = x^n + nx^{n-1}y+ \frac{n(n-1)}{2}x^{n-2}b^2 + ......+b^n[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
  3. Sep 29, 2017 #2

    JBD

    User Avatar

    Sir, in the generalization part, I have found some sources where the terms are infinite and not finite as written here. Can you post a link where I can find this formula? Thanks
     
  4. Sep 29, 2017 #3

    Svein

    User Avatar
    Science Advisor

    If n is not an integer, the binomial expansion will generally not stop. The reason why it stops when n is an integer - look at the (n+1)'th coefficient: [itex] \frac{n\cdot (n-1)\cdot ... \cdot 1}{1\cdot 2 \cdot ... \cdot n}[/itex]. The next one - if n is an integer - will be [itex] \frac{n\cdot (n-1)\cdot ... \cdot 1\cdot 0}{1\cdot 2 \cdot ... \cdot n\cdot (n+1)}=0[/itex].Thus [itex](x+y)^{\alpha} [/itex] with [itex]0<\alpha <1 [/itex] will start out as [itex]x^{\alpha}+\alpha x^{\alpha-1}y+... [/itex] where all exponents of x from #2 on will be negative.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: What is binomial theorem
  1. Binomial Theorem (Replies: 1)

  2. The Binomial Theorem (Replies: 2)

  3. Binomial Theorem (Replies: 6)

  4. Binomial Theorem (Replies: 3)

  5. Binomial theorem (Replies: 3)

Loading...