Algebra: Binomial Coefficient
Binomial theorem and sigma notation
Using binomial coefficients, we can write down the following formula for the #n#th power of a binomial:
Binomial theorem \[(a+b)^n = \binom{n}{0}\!a^n+\binom{n}{1}\!a^{n-1}b+\cdots + \binom{n}{n-1}\!ab^{n-1}+\binom{n}{n}\!b^n\]
The terms are of the form \(\binom{n}{k}\!a^{n-k}b^k\), with values for \(k\) from \(0,1,2,\ldots,n\).
In Pascal's triangle we have found that the binomial coefficient \(\binom{n}{k}\) is the #(k+1)#th number on the #(n+1)#th row of the triangle. That number arises from the coefficient of #a^kb^{n-k}# after working out the brackets of #(a+b)^n#. The coefficient of #a^kb^{n-k}# is \(\binom{n}{k}\).
Sigma notation
In the binomial theorem all terms of the form \(\binom{n}{k}\!a^{n-k}b^k\) are summated, where \(k\) ranges from \(0\) to \(n\). In order to write the sum efficiently we use the sigma notation: derived from the the Greek capital #\Sigma# we use \(\phantom{B}\sum\phantom{B}\) pronounced as "sigma" with \(k\) as a summation index. The binomial theorem is written as: \[(a+b)^n = \sum_{k=0}^n\binom{n}{k}\!a^{n-k}b^k\]
We can give the meaning of \[(a+b)^n = \sum_{k=0}^n\binom{n}{k}\!a^{n-k}b^k\]more precisely. The first value #0# of #k# is shown below the sigma and the last value #n# of #k# is shown above the sigma. Instead of #n# we could have written #k=n#. The summands are obtained by filling in all the integer values of #k# between #0# and #n# in the expression #\binom{n}{k}\!a^{n-k}b^k# behind the symbol #\sum#.
Using the sigma notation we can also specify infinite summations: \[\sum_{k=1}^{\infty}\frac{1}{k^2}= \frac{1}{1^2}+\frac{1}{2^2}+ \frac{1}{3^2}+ \frac{1}{4^2}+\cdots = \frac{\pi^2}{6}\] The chapter sequences and series will discuss this in more detail.
The sigma notation for a sum is often used in mathematics and statistics.
Substitute \(a=b=1\) in Newton's binomial : \[2^n=(1+1)^n = \sum_{k=0}^n\binom{n}{k}\!1^{nk}1^k=\sum_{k=0}^n\binom{n}{k}\]
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