Question

Use the binomial series to approximate up to and including terms of order 4. For which values of does this series converge?

Answer 1

The binomial series is expanded using the binomial theorem up to the fourth order, and converges if |b/a| < 1 for non-integer n, or for all real numbers if n is a non-negative integer.

Step-by-step explanation:

To approximate an expression using the binomial series, we apply the binomial theorem. The theorem enables us to expand expressions of the form (a + b)n into a series. An important condition for the convergence of this series when n is a non-integer is that |b/a| < 1. We are asked to approximate up to, and including, terms of order 4.

The binomial theorem is given as:

• (a + b)n = an + n*an-1b + n*(n-1)/2! * an-2b2 + n*(n-1)*(n-2)/3! * an-3b3 + ...

To illustrate, let's approximate (1 + x)n up to terms of the 4th order:

1. The term an is just 1.
2. The next term n*x represents the first order.
3. The term n*(n-1)/2! * x2 represents the second order.
4. Third order is given by n*(n-1)*(n-2)/3! * x3.
5. Finally, the fourth order term is n*(n-1)*(n-2)*(n-3)/4! * x4.

This series converges for all real numbers if n is a non-negative integer. For non-integral n, as mentioned previously, it converges when |x| < 1.