Section 23.1
Review: Power Series
Before solving DEs with series, we need to recall how infinite polynomials behave.
1
Definition
A power series centered at is:
2
Radius of Convergence
The Ratio Test
To find , we use:
.
This simplifies to .
3
Operations
Within the radius of convergence, we can differentiate term-by-term.
Note: The index shifts from to because the constant term disappears.
4
Worked Examples
Example 1: Geometric Series
Find the radius of convergence for .
1. Ratio Test:
.
Converges if .
. Interval: (-1, 1).
Example 2: Factorial
Find radius for .
.
is always true.
. Converges for all x.
Example 3: Index Shifting
Rewrite to start at .
Let .
When .
.
This skill is crucial for solving DEs.
5
Practice Quiz
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