Series, Convergence Tests and Taylor Polynomials

Unit 10, Infinite Sequences and Series, exists only in AP Calculus BC, and it carries a free response question almost every year. The unit rewards a clear decision process more than clever tricks.

Choosing a convergence test

Two warnings graders check every year:

• If lim aₙ = 0, the nth term test says nothing. Σ 1/n has terms tending to 0 and still diverges.
• A ratio test limit of exactly 1 is inconclusive. Switch tests.

Absolute and conditional convergence

A series converges absolutely if Σ |aₙ| converges. It converges conditionally if it converges but Σ |aₙ| does not. Σ (−1)ⁿ⁺¹/n is the standard example of conditional convergence.

Interval of convergence

1. Apply the ratio test to the power series and solve the inequality for x.
2. The ratio test says nothing at the endpoints, so test each endpoint separately.

Worked example: Σ (x − 2)ⁿ / (n · 3ⁿ) for n ≥ 1.

1. Ratio: |x − 2|/3 × n/(n + 1) tends to |x − 2|/3, so the series converges when |x − 2| < 3, that is −1 < x < 5.
2. At x = 5 the series is Σ 1/n, which diverges.
3. At x = −1 it is Σ (−1)ⁿ/n, which converges by the alternating series test.
4. Interval of convergence: [−1, 5). Radius: 3.

Taylor and Maclaurin series

f(x) = Σ f⁽ⁿ⁾(a)/n! × (x − a)ⁿ

A Maclaurin series is the case a = 0. Know these four without deriving them:

• eˣ = 1 + x + x²/2! + x³/3! + ... for all x
• sin x = x − x³/3! + x⁵/5! − ... for all x
• cos x = 1 − x²/2! + x⁴/4! − ... for all x
• 1/(1 − x) = 1 + x + x² + x³ + ... for |x| < 1

New series come from these by substitution, differentiation and integration. For example, replacing x with −x² in the last line gives 1/(1 + x²), and integrating that gives the series for arctan x.

The two error bounds

• Alternating series error bound: when the series alternates and its terms decrease to 0, the error after stopping is at most the first omitted term.
• Lagrange error bound: for a Taylor polynomial of degree n about a,

> |Rₙ(x)| ≤ M |x − a|ⁿ⁺¹ / (n + 1)!

where M bounds |f⁽ⁿ⁺¹⁾| between a and x.

The free response approach

Name the test you use and show that its conditions hold. For an interval of convergence, write the endpoint checks explicitly. For an error bound, say which bound you use and why it applies.