1. Apply the Ratio Test. 2. Solve the inequality L < 1 for |x - r|. 3. The radius of convergence, R, is the value such that |x - r| < R.

1. Find the radius of convergence, R. 2. Test the endpoints x = r - R and x = r + R in the original series. 3. Determine whether each endpoint converges or diverges. 4. Write the interval of convergence using appropriate brackets.

Apply the Ratio Test to the series.

Set the limit L < 1 and solve for |x - r|. The value on the right side of the inequality is the radius of convergence.

Test the endpoints of the interval by plugging them back into the original series and determining if they converge or diverge.

Substitute each endpoint value into the original power series. Then, analyze the resulting series using convergence tests (e.g., alternating series test, p-series test).

For a series $\sum a_n$, let $L=|limlimits_{n→ \infty}\frac{a_{n+1}}{a_n}|$. If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test is indeterminate.

$\sum^{\infty}_{n=0}a_n(x-r)^n$

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

$L < 1$

$L > 1$