EXAMPLE 9 Find the Maclaurin series for the function f(x) = 25 - x and its radius of convergence. SOLUTION We write f(x) in a form where we can use the binomial series. 25 . X 25 25 1 = -1/2 Using the binomial series with k = and with x replaced by -x/25, we have -1/2 8 V25 - x = N 1 = - 3 ( 1 + ( - 2)(- 25 ) + * 2 2! 25 + ( - 2) ( - 2)10 3 25 F . . 3! + (- 2)(-2)(-2) ... ( - 2 - n+1) X 25 + .. . n! = 1 + 1 + 1 . 3 2 +- 1 . 3 . 5 3 + . . . 21502 3!503 + - 1 . 3 . 5 . . . . . (2n - Dan + . . . n!507 We know from The Binomial Series that this series converges when |-x/25|