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SEQUENCES AND SERIES Section 12-5 Sigma Notation and the nth term SIGMA NOTATION OF A SERIES For any sequence a1, a2, a3 … , th e su m of th e first k term s m ay be w ritten w h ich is read “th e k n 1 a n summation from n=1 to k of a . Thus, k n 1 a n = a1 + a2 + a3 + … + ak where k is an integer value EXAMPLE #1 Write each expression in expanded form and then 7 find the sum. 1n b 2( ) (3 1) 1. 2. n 1 3 b 4 7 (3 1)= 82+244+730+2188=3244 b b 4 1n n 1 2(3 ) = 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + … . This is an infinite geometric series r=1/3 a1=2/3 so S = 2/3 / 1-1/3 = 1 EXAMPLE #2 E xp ress th e series 16+ 19+ 22+ 25+ … + 61 u sin g Sigma notation. I notice that each term is three more than the previous term. So I can write it as n+3. Since the first term is 16, n starts at 13. 5 8 n3 n 1 3 Another way I can look at it is would be looking for a pattern. Notice 3(5) + 1 = 16 3(6) + 1 = 19 3(7)+1=22 20 3n 1 n 5 N FACTORIAL The expression n! (n factorial) is defined as follows for n, an integer greater than zero. n! = n(n-1)(n-2)(n-3)… 1 EXAMPLE # 3 Express the series -4/1 + 16/2 - 64/6 + 256/24 in sigma notation. Notice the signs alternate beginning with a n negative sign. This indicates (-1) . We also notice that the numerator is powers of 4 and the denominator is n! So we have n 4 n ( 1) n1 n! 4 HW # 43 Section 12-5 Pp. 798-800 #14-35 all, 47a.,54