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Series

Series

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Calculus
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CALCULUS BC SERIES SERIES: PARTIAL SUMS: If (that it, the sequence of partial sums converges), then is the sum of the series so . In this case is CONVERGENT and has sum s. Thus a series converges if its sequence of partial sums converges. [Sec 11.2: p 2] EXAMPLES Does the series converge or diverge? 1. Partial sums: so, and thus the series converges and has sum = 1 2. Harmonic series: Partial sums: Thus the series diverges. [Sec 11.2: p3] 3. Here, (using partial fractions) Partial sum: Thus the series converges and has sum = 5. This is an example of a TELESCOPING SERIES.

Sequences

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CALCULUS BC NOTES: SEQ 11.1 (revised 2010) SEQUENCE: a list of numbers OR a function whose domain is the set of positive integers FIND FIRST 5 TERMS OF EACH SEQUENCE: 1) 2) 3) LIMIT of a SEQUENCE: (p 695) If where L is a finite number, the sequence CONVERGES. If no limit, the sequence DIVERGES. THEOREM: If (p 696) [Sec 11.1: p 2] FIND THE LIMIT OF EACH SEQUENCE. TELL WHETHER SEQUENCE CONVERGES OR DIVERGES. 4) 5) INCREASING/DECREASING SEQUENCES: increases if for all n decreases if for all n BOUNDED SEQUENCE : (p 700) Above: for all Below: for all [Sec 11.1: p 3] MONOTONIC SEQUENCE: always increasing or always decreasing
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