1 power series

A power series centered at x = a is: \sum a_n(x - a)^n = a_0 + a_1(x-a) + a_2(x - a)^2 + \cdots

example: Geometric series centered at x = 0 then 1 + x + x^2 + \cdots = \frac{1}{1-x} are convergent at -1 < x < 1

1.1 common power series

1.2 convergence theorem

There are 3 possibilities for a power series with respect to convergence

• There is a positive number R such that the series diverges for |x-a| > R but converges for all x such that |x-a| < R. The series may or may not converge at x = a \pm R
• The series converges for all x (R = \infty)
• The series converges for x = a and diverges elsewhere (R = 0)

1.3 interval of convergence

The previously mentioned R is called the radius of convergence

1.3.1 finding the interval of convergence

1. Use ratio or n-th root test to find absolute convergence
2. Test convergence at the endpoints using DCT, LCT, or AST

1.4 term-by-term differentiation

Inside of the interval of convergence, the series have derivatives of all orders.