(Infinite) Series Series a n = a 1 + a 2 + a a n +...

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1 (Infinite) Series Series a n = a 1 + a 2 + a a n +... What does it mean to add infinitely many terms? The sequence of partial sums S 1, S 2, S 3, S 4,...,S n,...,where nx S n = a i = a 1 + a 2 + a a n i=1 keeps track of the results of adding up more and more of the given terms a 1, a 2, a 3,... (in that order). Convergence, Divergence and Divergence to infinity for the series a n refer to those for the sequence S n.forconvergence, a n = lim i=1 nx a n = lim S n = S. The definition asks what happens to the sum in the long run if we add up more and more of the given terms a 1, a 2, a 3,... (in that order). The infinite geometric series (with a > 0) ar n 1 = a + ar + ar = a 1 r if r < 1, =+1 if r 1, and diverges if r apple 1. University Calculus II (University of Calgary) Winter / 5

2 Convergence/Divergence of the Sequence of Partial Sums in relation to the sequence of adjustments : We are interested in determining whether the sequence S n of partial sums converges or not. We can think of S 1, S 2, S 3, S 4,... as generated using a recursive process S n = S n 1 + a n for all n 2, and S 1 = a 1. We may view a n as the sequence of adjustment terms to update the value of S n from one partial sum to the next, from the (n 1)-th partial sum S n 1 to the n-th partial sum S n. Theorem (A Necessary Condition for Convergence) If P a n converges, then lim a n =0. Theorem (A Su cient Condition for Divergence, aka the n-th Term Test) If lim a n either does not exist or exists but is not equal to zero, then P a n diverges. These two Theorems tell us that: When we keep adding more and more numbers together, we can only hope the partial sums will get closer and closer to a limit when what we add to the sum during the successive stages eventually become arbitrarily small. If what we add to the sum during the successive stages do not eventually become arbitrarily small, then the partial sums will not be getting closer and closer to a limit. For a geometric series P ar n 1 (with a 6= 0), Since the sequence r n 1 either diverges or converges to 1 when r 1, the series P ar n 1 diverges. We know that ar n 1 = a 1 r when r < 1. Our Theorem implies that r n! 0asn!1,whichis consistently with what we know is true. University Calculus II (University of Calgary) Winter / 5

3 Notice that lim a n =0doesnotimplythat P a n converges. Convergence does not only require the adjustment term a n to become eventually arbitrarily small. It requires the adjustment term a n to become eventually arbitrarily small su The following series diverge to 1 since a n does not converge to 0 fast enough: ciently fast The last example is an important result: 1 n =+1. The sequence of partial sums grows very slowly, since the adjustment term 1 decreases to zero very n slowly. S 1 =1, S 10 = , S 100 = , S 1000 = , S = It does not seem that the partial sum will eventually surpass any positive number, no matter how large. But it does! University Calculus II (University of Calgary) Winter / 5

4 Properties of Convergence If P a n and P b n converge, and if k is any real number, then (a n + b n )= a n + b n, (ka n )=k a n Adding, removing or changing a finite number of terms in a series does not change whether the series converges or diverges, although it might change the value of the sum if the series converges. If N is any positive integer, then a n converges if and only if a n converges. Furthermore, n=n+1 a n = NX a n + a n. n=n+1 University Calculus II (University of Calgary) Winter / 5

5 Further s and Exercises Sum the series 1. Find an explicit expression for the partial sum S n and use it to find the sum of the series 1 n(n +1) = [Calculate the first few terms. Guess a formula. Then prove it by mathematical induction.] Find an explicit expression for the partial sum S n and use it to find the sum of the series ln 1+ 1 =ln ln ln n Determine if the following series converge. Find the sums for those that converge. 4 3 n 3,, e n 2 n +( 3) n, n 4 4 n Express the number as the quotient of two positive integers m n. What conclusions can we draw when we apply the n-th term test to the following series. n 3n +2, ( 1) n 1 1, n, 1 n, 1 2 n University Calculus II (University of Calgary) Winter / 5