Solution of Problem 1, Section 13, Mathematical Method in the Physical Sciences 3th - Mary L. Boas

Solution of Problem 1, Section 13

Mathematical Method in the Physical Sciences 3th - Mary L. Boas

Question:

Use the ratio test to show that a binomial series converges for |x| < 1.

Answer:

The ratio test is a method used to determine the convergence of a series. Given a series with terms a_n, the ratio test states that if the limit of |a(n+1) / an | as n approaches infinity is less than 1, then the series converges.

Consider the binomial series:

(1 + x)^m = Σ_{n=0}^∞ ( m choose n) x^n

where (m choose n) = m! / (n! (m-n)!)

Taking the ratio of consecutive terms, we have:

|a(n+1) / an| = | (m choose n+1) x^(n+1) /  (m choose n) x^n| 

= | (m-n) x / (n+1)|

= |x| (m-n) / (n+1)

As n approaches infinity, (m-n) approaches m and (n+1) apporaches infinity. Since |x| < 1, we have that |x| * (m-n) / (n+1) approaches infinity.

Therefore, the limit of | a(n+1) / an| as n approaches infinity is less than 1. By the ratio test, binomial series converges for |x| < 1.