Calculus Volume 2 · free preview
Power Series and Functions
A power series sums powers of x minus a, converging on an interval centered at a whose radius the ratio test decides, and within that interval it defines a function with a geometric series as the prototype.
1. Power series and convergence
Power series and convergence
2. The geometric prototype
The geometric prototype
3. Finding radius mechanically
Finding radius mechanically
4. The function viewpoint
The function viewpoint
5. Radius definition
The radius of convergence R of a power series is which quantity?
6. Geometric radius
The geometric series sum of x^n converges on which interval?
7. Endpoint checking
After the ratio test gives |x| less than R, why must the endpoints be checked separately?
8. Radius zero meaning
A power series with radius of convergence R = 0 does what?
9. Radius infinity example
The exponential series sum of x^n/n! has radius of convergence equal to what?
10. Geometric sum function
The geometric series sums to 1/(1 - x) on (-1, 1). As a ratio, what is the denominator? Answer like 1 - x.
11. Ratio test inequality
The ratio test converges when the limit rho is less than what number? Answer with a number.
12. Endpoint x equals one for sum x^n over n
At x = 1 the series sum of x^n/n becomes which famous series? Answer with two words.
13. Substitution preserves radius
Replacing x by 2x in the geometric series gives sum of (2x)^n. What is the new radius of convergence?
14. Center location
A power series in (x - a) is centered at x equals what value? Answer with the single letter a.
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