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 - 5/3x) we can consider as sum of infinite geometric series with a₁ = x²/3 and

common ratio r = 5/3x. For convergance abs(r) < 1, - 1 < 5/3x < 1 and - 3/5 < x < 3/5 interval of convergence and radius of convergense is 3/5.

Series itself: f(x) = ∑₁^∞x²/3(5/3x)^{n - 1} = ∑₁^∞5^{n -1}x^{n + 1}/3ⁿ