We explore Newton’s Binomial Theorem.

In this section, we extend the definition of to allow to be any real number and to be negative. First, we define to be zero if is negative. If is not a natural number, then we use instead of and we write . To define this, recall that The numerator of the last fraction contains factors, from counting down by one to . The computation of this numerator in no way requires to be a natural number. Hence, we define where is any real number and is any non-negative integer. If is a negative integer, we simply define to be zero and if , then we define to be one.

Newton’s Binomial Theorem involves powers of a binomial which are not whole numbers, like . Stating the theorem requires our new binomial coefficients, .

First, we write Now we use Newton’s Binomial Theorem with and . We have To make an approximation, we truncate the sum at some index. For the purpose of this example, we will truncate the series at . We have Since the infinite series in our approximation begins to alternate after the second term, the error in our four term approximation is less than twice the absolute value of the next term (since our estimate involves multiplying the series by 2). This next term corresponds to and its value is Hence, the error is at most Moreover, since the term corresponding to is negative, our approximation of as is an overestimate.

a) Approximate by writing it as . Truncate your infinite series at .

Approximation:

b) Approximate: by writing it as . Truncate your infinite series at .

Approximation:

c) Approximate by writing it as . Truncate your infinite series at .

Approximation:

d) Approximate by writing it as . Truncate your infinite series at .

Approximation:

- Proof
- We prove the case that and leave for the reader to prove the cases
and .

We compute the right hand side: since which agrees with the last factor in the numerator.