Lecture 29 – Power Series Def: The power series centered at x = a: x is the variable and the c’s are constants (coefficients)

For any power series, exactly one of the following is true:

Example 1 – Radius and Interval of Convergence Ratio Test: Series converges for

Example 2 – Radius and Interval of Convergence Ratio Test: Series converges for

Example 3 – Radius and Interval of Convergence Ratio Test:

Example 3 – continued (testing endpoints)

Example 4 – Radius and Interval of Convergence Root Test:

Example 4 – continued (testing endpoints)

Example 5 – Radius and Interval of Convergence Geometric Series:

Example 5 – continued – what is the converging value? Geometric Series:

Lecture 30 – More Power Series The geometric series: As a power series with a = 1, r = x and cn = 1 for all n: In other words, the function f(x) can be written as a power series. with

Example 1 Create new power series for other functions through: and sum, difference, multiplication, division, composition and differentiation and integration Example 1 with

Example 2 with

Consider the graphs:

Example 3 with

Need to solve for C. Set x = 0 to get: Test endpoints???

Example 4 with

Need to solve for C. Set x = 0 to get: Test endpoints???