find the power series and the interval of convergence

Question

use that fact that 1/1-x= sumx^n with interval of convergence (-1,1)

Answer 1

Use a substitution here with z = 4x². Then x²/(1 - 4x²) → 1/4 · [1 / (1 - z)] which converges to 1/4 · ∑ zⁿ =

1/4 · ∑ (4ⁿx²ⁿ) = ∑ (4^n-1 · x²ⁿ) = x² + 4x⁴ + 16x⁶ + ... provided -1 < 4x² < 1 so ...

-1/2 < x < 1/2 which represents the interval of convergence.