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Functions as Power Series
Recall the form of a power series: ; radius of convergence
- Converges if and diverges if .
- If , we know nothing.
Key Idea:
Differentiation
If , then .
will have the same radius of convergence as
Note: if , then the index starts at 1
Integration
If , then
will have the same radius of convergence as
Example
- radius of convergence is ∞ by ratio test:
Since , we can conclude that