11.4 Series & Sigma Notation

Note: a sequence or series does not have to be arithmetic or geometric Series = the terms of a sequence added together Arithmetic sequence: Note: a sequence or series does not have to be arithmetic or geometric 1, 3, 5, 7, 9, 11 Arithmetic series: 1 + 3 + 5 + 7 + 9 + 11 Geometric sequence: could be neither!! 5, 15, 45, 135 Geometric series: 5 + 15 + 45 + 135

Sigma Notation New notation: ∑  sigma = the sum of 1 + 3 + 5 + 7 + 9 + 11 general term? tn = 1 + (n – 1)(2) tn = 1 + 2n – 2 = 2n – 1 general term upper limit  = 1 + 3 + 5 + 7 + 9 + 11 lower limit 

Example 1 Evens: 2 + 4 + 6 + 8 + … + 100 general term? tn = 2 + (n – 1)(2) tn = 2 + 2n – 2 = 2n Find upper limit: (what term # is 100) n is term # tn = 2n 100 = 2n 50 = n = 2(1) + 2(2) + 2(3) + 2(4) + …+ 2(50)

Ex 2) Write the series in expanded form any letter  = (-1)1(1+2) + (-1)2(2+2) + (-1)3(3+2) + … + (-1)20(20+2) = (-1)(3) + (1)(4) + (-1)(5) + … + (1)(22) = – 3 + 4 – 5 + … + 22

Ex 3) Use sigma notation to write the series 10 + 15 + 20 + … + 100 Arithmetic d = 5 t1 = 10 general term? tn = 10 + (n – 1)(5) tn = 10 + 5n – 5 = 5n + 5 Find upper limit: (what term # is 100) tn = 5n + 5 100 = 5n + 5  19 = n or don’t forget parentheses!!

Ex 3) Use sigma notation to write the series series doesn’t end! (infinite) Patterns! Neither arithmetic nor geometric!! numer: 5 denom: evens (2n) alt. signs: (-1)n+1 (since 1st term is positive) infinity since series doesn’t end If 1st term is negative, use (-1)n Check it!!!

Homework #1105 Pg. 521 1 – 23 odd