series_intro.pdf - \u00a711.2 Series \u00a711.2(a What is a Series \u2022 Motivating question What is the sum of an infinite set of numbers \u2022 Ex 1 Let\u2019s try

# series_intro.pdf - u00a711.2 Series u00a711.2(a What is a...

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§11.2: Series§11.2(a): What is a Series?Motivating question: What is the sum of an infinite set of numbers?Ex. 1: Let’s try “adding up all the numbers” in a sequence.a){n}={1,2,3,4, . . .}While we cannot actually sit here and add infinitely many numbers together, wecertainly can add the firstNterms to get theNthpartial sum(sN), and lookfor a pattern in those partial sums:N= 1 :s1= 1N= 2 :s2= 1 + 2 = 3N= 3 :s3= 1 + 2 + 3 = 6...N= 100 :s100= 1 + 2 +. . .100 = 5050Conclusion:b)12n=12,14,18, . . .N= 1 :s1=12N= 2 :s2=12+14= 0.75N= 3 :s3=12+14+18= 0.875...N= 10 :s10=12+14+. . .1210= 0.99902N= 25 :s25=12+14+. . .1225= 0.99999997Conclusion:
c){(-1)n}={-1,1,-1,1, . . .}N= 1 :s1=-1N= 2 :s2=-1 + 1 = 0N= 3 :s3=-1 + 1-1 =-1N= 4 :s4=-1 + 1-1 + 1 = 0...Conclusion:Defn: The expressiona1+a2+. . . an+. . .we get when adding the terms of an infinite sequence{an}is called aninfinite series.It is denoted by eitherXn=1anorXan.If the infinite sum results in a finite number, we say the series.Otherwise, we say the series§11.2(b): Formalizing the Notion of Series ConvergenceDefn: Given a seriesXn=1an, letsNdenote theNthpartial sum:sN=NXn=1an=The series is said toconvergeiflimN→∞sN=sexists as a finite number. The numbersis called thesumof the series, and we write:s=Xn=1an.