AP Calculus BC: Infinite Sequences and Series

This chapter extends the concepts of calculus from finite processes to infinite ones. It begins with sequences, which are ordered lists of numbers, and investigates their limiting behavior. The core of the chapter, however, is on infinite series, which are the sums of the terms in a sequence. The central question for any series is whether it converges to a finite sum or diverges to infinity. You will learn to determine this for various types of series.

The main tools for this analysis are a collection of convergence tests. These include the Integral Test, p-Series Test, Direct and Limit Comparison Tests, Alternating Series Test, and the Ratio and Root Tests. A significant challenge for students is learning to quickly identify which test is most appropriate for a given series. The chapter also distinguishes between absolute and conditional convergence.

The final part of the chapter uses these tools to represent functions as infinite polynomials, known as power series. You will learn to construct Taylor and Maclaurin series for functions, determine their interval and radius of convergence, and use Taylor polynomials to approximate function values. A key skill is using the Lagrange Error Bound to determine the accuracy of these approximations. The entire chapter relies heavily on a strong foundation in limits and integration techniques.

Formulas for AP Calculus BC: Infinite Sequences and Series

Sum of a Convergent Geometric Series: $S = \frac{a}{1-r}$ Valid when the common ratio |r| < 1. Here 'a' is the first term.
Condition for p-Series Convergence: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ This series converges if p > 1 and diverges if p ≤ 1.
Limit for the Ratio Test: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ The series converges if L < 1, diverges if L > 1. The test is inconclusive if L = 1.
Limit for the Root Test: $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$ The series converges if L < 1, diverges if L > 1. The test is inconclusive if L = 1.
Alternating Series Error Bound: $|R_n| = |S - S_n| \le a_{n+1}$ The error in using the nth partial sum (S_n) is less than or equal to the first neglected term.
Taylor Series Expansion about x=a: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ f^(n)(a) is the nth derivative of f evaluated at a.
Maclaurin Series Expansion: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$ A special case of the Taylor series, centered at a=0.
Lagrange Error Bound for Taylor Polynomials: $|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$ M is the maximum value of |f^(n+1)(z)| on the interval between a and x.
Maclaurin Series for e^x: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ Converges for all x.
Maclaurin Series for sin(x): $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ Converges for all x.
Maclaurin Series for cos(x): $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ Converges for all x.
Maclaurin Series for 1/(1-x): $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots$ The basic geometric series. Converges for |x| < 1.