Rucete ✏ AP Calculus In a Nutshell
10. Sequences and Series
This chapter introduces infinite sequences and series (BC only), focusing on convergence tests, power series, Taylor and Maclaurin series, and error estimates.
A. Sequences of Real Numbers
• A sequence is a function an defined on the positive integers
• It converges to a number L if limn→∞ an = L If it has no finite limit, it diverges
Examples:
– an = 1/n → converges to 0 – an = (1 + 1/n)n → converges to e – an = (–1)n + 1 → diverges (oscillates)
B. Infinite Series
B1. Definitions
• An infinite series is a sum: ∑ an
• A geometric series: a + ar + ar² + ... Converges if |r| < 1; sum is a / (1 – r)
• Harmonic series ∑1/n diverges • p-series ∑1/nᵖ converges if p > 1
B2. Convergence Theorems
• If ∑an converges → an → 0 (Converse not true!)
• Finite terms can be added or removed without affecting convergence
• Multiplying all terms by a constant preserves convergence/divergence
B3. Tests for Convergence
Nth Term Test: If lim an ≠ 0 → series diverges (If lim = 0, test is inconclusive)
Geometric Series: Converges if |r| < 1
Integral Test: If f is positive, continuous, decreasing and f(n) = an: ∑an converges ⇔ ∫f(x)dx converges
p-Series: ∑1/nᵖ converges if p > 1, diverges if p ≤ 1
Comparison Test: Compare an with known convergent or divergent series – If an ≤ bn and ∑bn converges → ∑an converges – If an ≥ bn and ∑bn diverges → ∑an diverges
Limit Comparison Test: lim an / bn = L (finite, positive) → both converge/diverge together
Ratio Test: lim |an+1 / an| = L – If L < 1 → converges – If L > 1 → diverges – If L = 1 → inconclusive
Root Test: lim √[n]{|an|} = L – Same rule as Ratio Test – Not on AP Exam but useful
B5. Alternating Series
Alternating Series Test: If an is decreasing and → 0, ∑(–1)nan converges
• If ∑|an| converges → absolutely convergent • If ∑an converges but ∑|an| diverges → conditionally convergent
Alternating Series Error Bound: |Rn| < an+1 → The error is less than the first omitted term
C. Power Series
• A power series is of the form: ∑ cn(x – a)n
• Centered at x = a • May converge: – Only at x = a – For |x – a| < R (interval of convergence) – For all x
Radius of Convergence (R):
• Use Ratio or Root Test • Interval of convergence (IOC) = (a – R, a + R), check endpoints separately
D. Taylor and Maclaurin Series
• Taylor Series for f(x) at x = a:
f(x) = ∑ f⁽ⁿ⁾(a) / n! · (x – a)n
• Maclaurin Series is Taylor series centered at a = 0:
f(x) = ∑ f⁽ⁿ⁾(0) / n! · xⁿ
Common Maclaurin Series:
• eˣ = ∑ xⁿ / n! for all x
• sin x = ∑ (–1)ⁿ x²ⁿ⁺¹ / (2n + 1)! for all x
• cos x = ∑ (–1)ⁿ x²ⁿ / (2n)! for all x
• 1 / (1 – x) = ∑ xⁿ |x| < 1
• ln(1 + x) = ∑ (–1)ⁿ⁺¹ xⁿ / n |x| < 1
• arctan x = ∑ (–1)ⁿ x²ⁿ⁺¹ / (2n + 1) |x| ≤ 1
E. Error Estimation (Lagrange Form)
• Taylor remainder (error): Rn(x) = f⁽ⁿ⁺¹⁾(z) / (n + 1)! · (x – a)n+1
• z is between a and x • Bound |Rn(x)| using max of |f⁽ⁿ⁺¹⁾(z)|
F. Using Series to Approximate Functions
• Truncate after n terms to estimate function values
Example: Approximate sin(1) using first 3 terms sin(1) ≈ 1 – 1³/3! + 1⁵/5! = 1 – 0.1667 + 0.0083 = 0.8416
G. Term-by-Term Differentiation and Integration
• If power series converges on interval I, then:
– It can be differentiated/integrated term-by-term – The new series has the same interval of convergence
Example: f(x) = ∑ xⁿ → f′(x) = ∑ n·xⁿ⁻¹ ∫f(x) dx = ∑ xⁿ⁺¹ / (n + 1)
In a Nutshell
Infinite series allow representation of complex functions as sums of polynomials. Key convergence tests ensure valid use. Power, Taylor, and Maclaurin series provide powerful tools for approximation and computation, with error bounds quantifying accuracy. BC students use these series to analyze functions and model behavior.