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).

A series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$, where n is a non-negative integer, $a_n$ is a sequence of real numbers, and r is a real number.

The radius of convergence defines the interval around the center of a power series where the series converges.

The interval of convergence is the set of all x-values for which a power series converges. It may include or exclude the endpoints.

It means the power series is expressed in terms of (x - r), where 'r' is the center of the interval of convergence.

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.