AP Calculus BC / Unit 10: Infinite Sequences and Series
10.13 Radius and Interval of Convergence of Power Series
Notes
A power series $\displaystyle \sum c_n(x-a)^n$ converges on an interval centered at $a$.
- Use the ratio test to find the radius $R$.
- Then check each endpoint separately.
Practice
The radius of convergence of $\displaystyle \sum\frac{x^n}{3^n}$:
Geometric with $\displaystyle r=\frac{x}{3}$; need $|x|<3$.
The interval of convergence of $\displaystyle \sum\frac{x^n}{n}$:
At $x=-1$ it's alternating harmonic (converges); at $x=1$ it's harmonic (diverges).
$\displaystyle \sum\frac{x^n}{n!}$ converges for:
The ratio is $\displaystyle \frac{|x|}{n+1}\to0$.
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