use a geometric series to represent .....

Question

use a geometric series to represent f(x)= (x²)/(3-5x) as a Maclaurin series. What is its radius of convergence of the series?

Answer 1

f(x) = x² / (3 - 5x) = (x²/3) / (1 - 5x/3) = a / (1 - r)

f(x) = ∑_{(from 0 to ∞)} arⁿ = ∑_{(from 0 to ∞)} (x²/3)(5x/3)ⁿ = (1/3)∑_{(from 0 to ∞)} [(5/3)ⁿxⁿ⁺²]

The series converges as long as -1 < r < 1. That is, when -1 < 5x/3 < 1.

So, -3/5 < x < 3/5. Radius of convergence = half the length of the interval of convergence = 3/5.