Convergence tests, Taylor and Maclaurin series, and sequence concepts tested on AP Calculus BC. Front: the term or test name. Back: definition or the condition it checks.
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- What is a sequence?
- An ordered list of numbers defined by a function whose domain is the positive integers, written as {a_n}.
- When does a sequence converge?
- When the limit of a_n as n approaches infinity equals a finite number L.
- When does a sequence diverge?
- When the limit of a_n does not exist or is infinite.
- What is a monotonically increasing sequence?
- A sequence where each term is greater than or equal to the previous term, a_n <= a_(n+1) for all n.
- What is a bounded sequence?
- A sequence where all terms lie between two fixed numbers, m <= a_n <= M for all n.
- What is a series?
- The sum of the terms of a sequence, written as the sum from n=1 to infinity of a_n.
- What is a convergent series?
- A series whose sequence of partial sums has a finite limit.
- What is a divergent series?
- A series whose sequence of partial sums either diverges to infinity, oscillates, or has no limit.
- What does the Divergence Test tell you?
- If the limit of a_n as n approaches infinity is not zero, then the series sum of a_n diverges. (If the limit is zero, the test is inconclusive.)
- What is a geometric series?
- A series of the form sum of ar^n from n=0 to infinity, where a is the first term and r is the common ratio.
- When does a geometric series converge?
- When the absolute value of the common ratio |r| is less than 1. The sum equals a / (1 - r).
- What is a p-series?
- A series of the form sum of 1 / n^p from n=1 to infinity, where p is a positive constant.
- When does a p-series converge?
- When p > 1. It diverges when p <= 1.
- What does the Integral Test state?
- If f is positive, continuous, and decreasing on [1, infinity), then the series sum of a_n and the integral of f(x) from 1 to infinity both converge or both diverge.
- What does the Comparison Test state?
- If 0 <= a_n <= b_n for all n and series b_n converges, then series a_n converges. If a_n >= b_n >= 0 and series b_n diverges, then series a_n diverges.