Educator

Power Series

Main definitions and pattern:

Definitions: A power series is a series of the form . (The c_n ’s are the coefficients, expressions that might involve n, but won’t involve x.)

Pattern: The power series always converges for values of x within some radius R around the center a. For a − R < x < a + R (i.e. |x − a| < R), it is Absolutely Convergent . For x < a − R or x > a + R (i.e. |x − a| > R), it Diverges . At the endpoints x = a − R and x = a + R, it might be conditionally convergent, absolutely convergent, or divergent.

R is called the radius of convergence. We can have R = 0 or R = ∞. The interval a − R < x < a + R (or a − R ≤ x < a + R, or a − R < x ≤ a + R, or a − R ≤ x ≤ a + R) is called the interval of convergence.

Hints and tips:

• For most power series, you can use the Ratio Test to find the radus of convergence.

• On a few examples, you should use the Root Test. These examples usually have the form (a function of n)ⁿ .

• However, you can never use the Ratio or Root Test to check the endpoints, since they will give you L = 1, which is inconclusive.

• You must always check each endpoint individually, using some test other than Ratio or Root. Common favorites are the Limit Comparison Test with a p-series, Alternating Series Test, and Test For Divergence.

• A factorial in the denominator often leads to R = ∞.

• Memorize the geometric series expansion for -1 < x < 1.

• You can often derive other power series from the geometric series by the following methods.

• Algebraic manipulations, e.g. multiplying by x. These won’t change the radius of convergence or whether the series converges at the endpoints.

• Substitutions, e.g. replacing x by 2x or x² . This will change the radius of convergence.

• Derivatives and integrals. These won’t change the radius of convergence, but they might change whether the series converges at the endpoints.

• Other common series that are worth memorizing (although they can be derived from the geometric series) are for −1 ≤ x < 1 and arctan for −1 ≤ x ≤ 1.