Find a power series representation and radius of convergence for f(x) =(x^4)*(25

Question

Find a power series representation and radius of convergence for f(x) =(x^4)*(25+x^2)^(-1/2)

Explanation / Answer

Start with the Binomial series with exponent -1/2: (1 + t)^(-1/2) = 1 + ?(n = 1 to ?) C(-1/2, n) t^n, where C(-1/2, n) = (-1/2)(-1/2 - 1)...(-1/2 - n + 1)/n! = (-1/2)(-3/2)...(-(2n-1)/2)/n! = (-1)^n (1 * 3 * ... * (2n+1))/(2^n * n!). Hence, 1/?(1 + t) = 1 + ?(n = 1 to ?) [(-1)^n (1 * 3 * ... * (2n+1))/(2^n * n!)] t^n. ------------- Let t = x^2/25: 1/?(1 + x^2/25) = 1 + ?(n = 1 to ?) [(-1)^n (1 * 3 * ... * (2n+1))/(2^n * n!)] (x^2/25)^n ==> 1/?((1/25) (25 + x^2)) = 1 + ?(n = 1 to ?) [(-1)^n (1 * 3 * ... * (2n+1))/(2^n * n!)] x^(2n)/25^n ==> 5/?(25 + x^2) = 1 + ?(n = 1 to ?) [(-1)^n (1 * 3 * ... * (2n+1))/(50^n * n!)] x^(2n). Multiply both sides by (1/5)x^4: x^4/?(25 + x^2) = 1 + ?(n = 1 to ?) [(-1)^n (1 * 3 * ... * (2n+1))/(5 * 50^n * n!)] x^(2n+4). -------------------- As for the radius of convergence, the basic binomial series converges (save possible endpoints) when |t| < 1. The only change to the radius of convergence above occurs when we let t = x^2/25. ==> |x^2/25| < 1 ==> |x| < 5; so the radius of convergence is 5.

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