Unit 10 (BC only) covers infinite series — representing functions as infinite sums. Taylor and Maclaurin series are among the most powerful concepts in calculus with wide applications in science and engineering.
nth term test: if aₙ ↛ 0, series diverges. Geometric: converges if |r| < 1, sum = a/(1-r). p-series: Σ1/nᵖ converges if p > 1. Integral test: compare to improper integral. Comparison: compare to known series. Limit comparison: lim aₙ/bₙ = finite positive → same behaviour. Ratio test: lim |aₙ₊₁/aₙ| < 1 → converges. Alternating series: decreasing terms → 0 → converges. Root test: lim ⁿ√|aₙ| < 1 → converges.
Σcₙ(x-a)ⁿ: converges for |x-a| < R (radius of convergence). Find R using ratio test. Interval of convergence: check endpoints separately. Can differentiate and integrate power series term by term within interval.
Taylor series of f about x = a: Σ f⁽ⁿ⁾(a)/n! (x-a)ⁿ. Maclaurin: Taylor at a = 0. Key series: eˣ = Σxⁿ/n!, sinx = Σ(-1)ⁿx²ⁿ⁺¹/(2n+1)!, cosx = Σ(-1)ⁿx²ⁿ/(2n)!, 1/(1-x) = Σxⁿ. Lagrange error bound: |Rₙ(x)| ≤ M|x-a|ⁿ⁺¹/(n+1)! where M = max|f⁽ⁿ⁺¹⁾| on interval. Alternating series error: |error| ≤ |first omitted term|.
Taylor series allow you to represent any smooth function as an infinite polynomial — and polynomials are easy to differentiate, integrate, and compute with. Applications: (1) Approximation: use first few terms for numerical calculations (how calculators compute sin, cos, eˣ). (2) Physics: simplify complex equations (small angle approximation: sinθ ≈ θ). (3) Differential equations: find series solutions when exact solutions are impossible. (4) Computing: many algorithms use Taylor polynomials for efficiency. The Lagrange error bound tells you exactly how accurate your approximation is with n terms.
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