Binomial Series
Let us consider the following example.
A defense contractor has two factories at two different places. The first factory
produces items at a rate of "a" items per year. The second factory produces items at
a rate of "b" items per year. The combined output rate at the end of the first year
is (a+b). From the second year onwards, the combined out put rate is multiplied by
(a+b) to the combined output rate of the previous year i.e. the combined output
rate at the end of the second year is (a+b)2, the combined output
rate at the end of the third year is (a+b)3 and so on. What is the combined output rate at the end of the nth year?
This can be found out using binomial formula expansion and can be written as
follows.
(a+b)2 = a2 + 2ab + b2
(a+b)3 = a3 + 3a2b + 3ab2 + b3
Similarly we can write for (a+b)4, (a+b)5 and generalize
(a+b)n using Binomial Theorem as follows.
(a+b)n = an + nan-1b + {n(n-1)/2}an-2b2 + ... +bn.
We can also represent the above expression using binomial coefficients as follows.
(a+b)n = C(n,0)an +C(n,1)an-1b + C(n,2)an-2b2+ ... +C(n,k)akbn-k+ ... +C(n,n)bn
where C(n,k) =n!/{(n-k)! k!}
Thus the binomial formula can be used to expand the terms of the n th
power of sum of two numbers.
Now we get Binomial Series if we rewrite the binomial in the form 1+x (by having
a=1 and b=x) as follows.
(1+x)n= 1+nx +{n(n-1)}x2/2! +{n(n-1)(n-2)}x3/3!+ ...+ x n.