Representing Functions by Power Series

A power series is said to represent a function f with a domain equal to the interval I of convergence of the series if the series converges to f(x) on that interval. That’s if:

Example

Theorem

Examples

Example(1)

We notice that And we know that:

Solution

Example(2)

We notice that And we know that:

Solution

Question What about the convergence at the end points?

1. The function ln(x-1) is not defined at x = 1 2. We can show easily that the series is convergent if x = -1 (how?) But does it converge to ln2? The answer to this question has to wait till we introduce Able’s Theorem

Approximating ln2

Example(3)

We notice that And we know that:

Solution

Question What about the convergence at the end points?

We can show easily that the series is convergent if x = 1or x = -1 (how?) But does it converge to arctan1 = π/4 & arctan(-1) = π/4 respectively ? The answer to this question has to wait until after we introduce Able’s Theorem !

Approximating arctan (0.5)

Showing that this series converges to e

Approximating e

Question Approximate 3 √e

Able’s Theorem

Home Quiz (2)

Homework