Calculus BC · Unit 10 · Paper 1

Infinite Sequences & Series unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 63 terms and is the same for everyone, so a teacher can assign “Unit 10, Paper 1” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

Alternating series error bound

A Taylor polynomial is finite

Taylor series

p-series

Maclaurin for 1/(1 − x)

Taylor polynomial construction

Radius zero or infinite

Power series centered at a

Conditional convergence

Absolute vs conditional convergence

Taylor series centered elsewhere

Common series to recognize instantly

Short answer 1. Define or explain: Substitution into a known series

3 pts

Short answer 2. Define or explain: Endpoints can disagree

3 pts

Short answer 3. Define or explain: Term-by-term differentiation and integration

3 pts

Short answer 4. Define or explain: Test-selection order

3 pts

Free response

9 pts

NO CALCULATOR. The Taylor series for a function f about x = 2 is given by Σ (from n = 1 to ∞) (x − 2)ⁿ / (n · 4ⁿ) = (x − 2)/4 + (x − 2)²/(2·16) + (x − 2)³/(3·64) + ⋯ and converges to f(x) on its interval of convergence.

A. Using the ratio test, find the interval of convergence of the Taylor series for f about x = 2. Justify your answer.

B. Find the first three nonzero terms and the general term of the Taylor series for f′, the derivative of f, about x = 2.

C. The Taylor series for f′ found in part B is a geometric series. Show that f′(x) = 1/(6 − x) for all x in the interval of convergence of that series.

D. Does the Taylor series for f′ converge at x = 8? Give a reason for your answer.