Question: Find a power series representation for the function and determine the interval of convergence of f(x) = 5/(1 - 4x

^{2})

f(x)

= 5/(1 - 4x

^{2})

= 5 * 1/(1 - 4x

^{2})

Therefore, the power series representation is

5 * summation (4x

^{2})

^{n}

Geometric series, so...

|r|< 1

|4x

^{2}| < 1

|x

^{2}| < 1/4

|x| < 1/2

Therefore, the radius of convergence is 1/2 and the interval of convergence is -1/2, 1/2

Check endpoints

x = -1/2: 5 summation 1 diverges

x = 1/2: 5 summation 1 diverges

Therefore, the interval of convergence is (-1/2, 1/2)