Section 23.2

Review: Taylor Series

Representing functions as infinite polynomials is the heart of the series method.

1

Definition

If has derivatives of all orders at :

2

Common Maclaurin Series

Must Memorize (Centered at 0)

  • Exponential:
  • Sine:
  • Cosine:
  • Geometric: (|x| < 1)
3

Analytic Functions

A function is analytic at if it equals its Taylor series in a neighborhood of .

For DEs , we need the coefficients to be analytic.

4

Worked Examples

Example 1: Finding Series

Find the Maclaurin series for .

Use the known series for :

.

Shift index if needed (): .

Example 2: Initial Conditions

If , express and in terms of .

.

.

.

Generally, .

Example 3: Visual

Approximating sine with higher degree polynomials.

5

Practice Quiz

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