Sequences and Series in Math 140: Definition, Summation Notation, and Recursive Formulas, Study notes of Algebra

Download Sequences and Series in Math 140: Definition, Summation Notation, and Recursive Formulas and more Study notes Algebra in PDF only on Docsity! Math 140 E2 Nov.21,2008 10.1 Sequences and Series 1. Sequence: A sequence is a function whose domain is the set of positive integers Example: 1, 1/2, 1/3, 1/5, · · · We usually use subscripted letters to represent the term of a sequence A sequence can be represented by listing the terms or using the formula for the nth term (general term) Example: for the sequence whose nth term is bn = (1/2)n Example: Finding the general term (1)e, e2/2, e3/2, · · · (2)1, 4, 9, 16, 25, · · · Example: Write the first six terms of the following sequence bn = {(−1)n+1( 2 n )} 2. Summation Notation Sn = n∑ i=1 ai = a1 + a2 + · · ·+ an Example:Write out each sums (1) 5∑ k=1 1 k (2) 6∑ k=0 (−1)k5k (3) n∑ k=1 (2k + 1) 3. Recursive Formulas: A second way to define a sequence is to assign a value to the first (or the first few) term(s) and specify the nth term by a formula that involves one or more of the terms preceding it. Example: Write down the first five terms of the following recursively defined sequence (1) a1 = 1, an = 4an−1 (2) a1 = 5, an+1 = 3an 1