# Thomas' Calculus: Early Transcendentals

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Overview of Sequences and SeriesMAT 104 – Frank Swenton, Summer 2000Sequencesare ordered lists of real numbers, such as “a1, a2, a3, . . .”, sometimes written{an}Just as with limits of functions on the real line, we can talk about limits of sequences; if the numbersin the sequence approach some real numberL, then the sequence has a limit. Otherwise, it mayapproachor-∞, or it may not approach anything at all in particular.Note thatiflimx→∞f(x) either exists or is infinite, then the limitlimn→∞f(n) is the same (the graph ofthe sequence lies on the graph of the function, so if the function approaches a limit, the sequenceis stuck to the same limit). This fact can be useful in computing limits of some sequences, sincelimits of functions can often be more easily evaluated by l’Hˆopital’s rule. For example:Using l’Hˆopital’s rule,limx→∞x2ex= limx→∞2xex= limx→∞2ex= 0, solimn→∞n2en= 0 as well.Some useful limits to know:If|r|<1, thenlimn→∞rn= 0limn→∞nn= 1(and the same fornn2,nn3, etc.)limn→∞nn! =Some limits that equale:limn→∞1 +1nn,limn→∞n+ 1nnSome limits that equal1e:limn→∞1-1nn,limn→∞n-1nn,limn→∞nn+ 1nRemember the hierarchy of functions (on the “limit comparison test” page);Iff(n)g(n), thenlimn→∞f(n)g(n)= 0, andlimn→∞g(n)f(n)=.For example,limn→∞100nn!= 0,limn→∞n1002n= 0, etc.Remember the rules of exponentiation:am+n=am·an,a-n=1an, andamn= (am)nSeries(sums of infinitely many terms)Given a sequencea1, a2, a3, . . ., we may want to add up all of the values, i.e.a1+a2+a3+· · ·.This is called aseries, and it is denotedXn=1an. The individual numbers being added together arecalled thetermsof the series. To add up an infinite number of terms, we first define thepartialsumsof the series,Sn=a1+a2+· · ·+an, i.e.Snis the sum of the firstnterms of the series.We then define the meaning of the infinite sum byXn=1an=limn→∞Sn. If this limit exists as a realnumber, we call the seriesconvergent; if the limit doesn’t exist, we call the seriesdivergent.

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