How far a power series reaches

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

What is the radius of convergence of the power series with terms x to the n over 2 to the n?
Answer: ______________
What is the radius of convergence of the series with terms x to the n over 2 to the n?
Answer: ______________
Differentiating a power series term by term keeps the same radius of convergence.
Circle one: True False
What is the radius of convergence of the series with terms n factorial times x to the n?
- 1
- Infinite
- 0
Integrating the geometric series with terms x to the n term by term gives a new series. What is its radius of convergence?
Answer: ______________
Differentiating the series with terms x to the n over n squared term by term gives terms x to the (n-1) over n. What is the new radius?
- 1, the same as before
- 0
- 2
- Infinite
What is the radius of convergence of the series with terms n factorial times x to the n?
- 0
- 1
- 1/2
- It converges for all x
The series with terms x to the n over n converges at x = minus 1 and diverges at x = 1. Its radius is 1. What is its interval of convergence?
- (-1, 1)
- [-1, 1)
- (-1, 1]
Sam says a radius of 1 means the series converges at both endpoints. Is Sam right?
Circle one: True False
What is the radius of convergence of the series with terms (2x) to the n over n?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

2 · The ratio of consecutive absolute terms is the absolute value of x over 2, below 1 exactly when x is within 2 of 0.
2 · The ratio of consecutive absolute terms is the absolute value of x over 2, which stays below 1 exactly within 2 of 0.
True · Term by term calculus preserves R. Only endpoint inclusion may change.
0 · The ratio grows past 1 for every nonzero x, so only the center itself converges.
1 · Term by term integration preserves the radius, and the geometric series has radius 1.
1, the same as before · Term by term differentiation preserves the radius of convergence; only the endpoints may change fate.
0 · The ratio grows like (n + 1) times the absolute value of x, which exceeds 1 for every nonzero x.
[-1, 1) · Radius 1 gives the bulk (minus 1, 1), then minus 1 stays while 1 goes.
False · The radius says nothing about the boundary. Each endpoint needs its own test.
0.5 · The ratio is 2 times the absolute value of x, which stays below 1 exactly within 0.5 of 0.