Question

Use the ratio test to find the radius of convergence and interval of convergence of the power series

x-x²/4+x³/9+x/⁴16+x⁵/25

Answer 1

The radius of convergence is infinite and the interval of convergence is (-∞, ∞).

Step-by-step explanation:

To find the radius of convergence and interval of convergence of the given power series,

Σ (x^n/n^2), we can use the ratio test.

1. Apply the ratio test: |x^(n+1)/(n+1)^2 * n^2/(x^n)|
2. Simplify the expression: |x/(n+1)^2|
3. Take the limit as n approaches infinity: lim(|x/(n+1)^2|) = 0

Since the limit is less than 1 for all x, the power series converges for all values of x. Therefore, the radius of convergence is infinite and the interval of convergence is (-∞, ∞).