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?
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An ordered list of numbers defined by a function whose domain is the positive integers, written as {a_n}.- When does a sequence converge?
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When the limit of a_n as n approaches infinity equals a finite number L.- When does a sequence diverge?
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When the limit of a_n does not exist or is infinite.- What is a monotonically increasing sequence?
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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?
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A sequence where all terms lie between two fixed numbers, m <= a_n <= M for all n.- What is a series?
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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?
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A series whose sequence of partial sums has a finite limit.- What is a divergent series?
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A series whose sequence of partial sums either diverges to infinity, oscillates, or has no limit.- What does the Divergence Test tell you?
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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?
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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?
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When the absolute value of the common ratio |r| is less than 1. The sum equals a / (1 - r).- What is a p-series?
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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?
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When p > 1. It diverges when p <= 1.- What does the Integral Test state?
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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?
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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.
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