Infinite sequences and series (BC) practice problems AP Calculus, Unit 10
BC only. Convergence tests, geometric and alternating series, Taylor and Maclaurin polynomials, and radius of convergence. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.
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Key equations for infinite sequences and series (bc)
| a/(1 − r), for |r| < 1 | sum of a geometric series |
|---|---|
| lim |an+1/an| < 1 | ratio test: converges |
| ex = Σ xn/n! | Maclaurin series for ex |
| |error| ≤ |first omitted term| | alternating series error bound |
Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.
Warm-up questions
Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.
Hint
Geometric with first term 1 and ratio 1/2.
Worked solution
- Sum = a/(1 − r) = 1/(1 − 1/2).
- = 2.
Hint
Compare with the integral of 1/x, or group the terms.
Worked solution
- It is a p-series with p = 1, and p-series converge only for p > 1.
- So it diverges, even though its terms go to 0: C.
Hint
The general term is xn/n!.
Worked solution
- The x3 term is x3/3!.
- 3! = 6, so the coefficient is 1/6.
Exam-style questions
Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.
Hint
Geometric with a = 3 and r = −1/2.
Worked solution
- |r| < 1, so the sum is 3/(1 + 1/2).
- = 2.
Hint
Find lim |an+1/an|.
Worked solution
- an+1/an = [(n + 1)/2n+1] × [2n/n] = (n + 1)/(2n).
- This tends to 1/2 < 1, so the series converges: B.
Hint
It is geometric with ratio x/3.
Worked solution
- It converges when |x/3| < 1.
- That is |x| < 3, so R = 3.
Hint
cos x ≈ 1 − x2/2.
Worked solution
- 1 − 0.22/2 = 1 − 0.02.
- = 0.98. (The true value is 0.98007.)
Hint
The error is at most the first omitted term.
Worked solution
- The first omitted term is the 4th: 1/42.
- Error ≤ 1/16 = 0.0625.
Free-response practice
One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.
No-calculator free-response style · 6 points · about 15 minutes
Let f(x) = e2x.
- (a) Write the first four nonzero terms of the Maclaurin series for f.
- (b) Write the general term.
- (c) Find the radius of convergence. Show your reasoning.
- (d) Use the first four terms to approximate e0.2.
Write your full answer on paper first, then open the scoring guide and tick each point you earned.
Scoring guide and model answer
Part (a)
Substitute 2x into eu: 1 + 2x + 2x2 + (4/3)x3 + …
Part (b)
(2x)n/n! = 2nxn/n!.
Part (c)
Ratio: |2x|/(n + 1) → 0 < 1 for every x, so the radius is infinite.
Part (d)
x = 0.1: 1 + 0.2 + 0.02 + 0.001333 ≈ 1.22133. (True value 1.22140.)
Your mark: 0 of 6
Common mistakes in infinite sequences and series (bc)
- Concluding a series converges just because its terms tend to 0.
- Forgetting to check the endpoints when finding an interval of convergence.
- Using n! wrongly in a Taylor coefficient: it is f(n)(a)/n!.
Keep going
Revise the ideas behind these questions:
Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units