Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 5 Notes These notes correspond to Section 8.2 in the text. Series, cont'd In the previous lecture, we defined the concept of an infinite series, and what it means for a series to converge to a finite sum, or to diverge. We also worked with one particular type of series, a geometric series, for which it is particularly easy to determine whether it converges, and to compute its limit when it does exist. Now, we consider other types of series and investigate their behavior. Telescoping Series Consider the series 1 X 1 1 − : n n + 1 n=1 If we write out the first few terms, we obtain 1 X 1 1 1 1 1 1 1 1 1 − = 1 − + − + − + − + ··· n n + 1 2 2 3 3 4 4 5 n=1 1 1 1 1 1 1 = 1 + − + − + − + ··· 2 2 3 3 4 4 = 1: We see that nearly all of the fractions cancel one another, revealing the limit. This is an example of a telescoping series. It turns out that many series have this property, even though it is not immediately obvious. Example The series 1 X 1 n(n + 2) n=1 is also a telescoping series. To see this, we compute the partial fraction decomposition of each term. This decomposition has the form 1 A B = + : n(n + 2) n n + 2 1 To compute A and B, we multipy both sides by the common denominator n(n + 2) and obtain 1 = A(n + 2) + Bn: Substituting n = 0 yields A = 1=2, and substituting n = −2 yields B = −1=2. The series is now 1 1 X 1 1 X 1 1 = − n(n + 2) 2 n n + 2 n=1 n=1 1 1 1 1 1 1 1 1 = 1 − + − + − + − + ··· 2 3 2 4 3 5 4 6 1 1 1 1 1 1 = 1 + − + − + + ··· 2 2 3 3 4 4 1 1 = 1 + 2 2 3 = : 4 2 Harmonic Series One series of interest is actually a divergent one, the harmonic series 1 X 1 : n n=1 This series is the best-known example of a series that diverges even though the sequence of its terms converges to zero. To see that it diverges, we can use the fact that the terms of this series are at least as large as those of the series 1 X 1 ; 2dlog2 ne n=1 where, for each n, the nth term is 1 divided by the smallest power of 2 that is greater than or equal to n. It can be shown that this series diverges by examining the sequence of partial sums directly, and since P 1=n has terms that are at least as large, it must diverge as well. Although this series diverges, its terms are quite close to that of a series that converges. In fact, the series 1 X 1 ; n1+ n=1 for any > 0, is convergent. 2 Basic Convergence Tests Because summing a series requires adding infinitely many numbers, it makes sense, intuitively, that these numbers must get smaller as the index n ! 1, if there is to be any hope that the sum will converge to a finite number. This is in fact the case: if a series converges, the sequence of its terms must converge to zero. However, the converse is not true: if a series has terms that converge to zero, it does not necessarily converge. The harmonic series, above, is an example of a divergent series whose terms converge to zero. Instead, we can use the contrapositive statement to arrive at a condition for divergence, rather than convergence: if the terms of a series do not converge to zero, then it diverges. We will learn about several tests that can be used to prove that a series converges, but for now, we note that certain simple combinations or modifications of convergent series are also convergent. Specifically, if 1 1 X X an and bn n=1 n=1 are convergent series, with limits Sa and Sb, respectively, then the series 1 1 1 X X X an + bn; an − bn; can; n=1 n=1 n=1 where c is a constant, are also convergent, with limits 1 1 1 X X X an + bn = Sa + Sb; an − bn = Sa − Sb; can = cSa: n=1 n=1 n=1 Example Using the result of previous examples, we have 1 1 1 X 1 1 X 1 X 1 + = + 2n n(n + 2) 2n n(n + 2) n=1 n=1 n=1 1 1 X 1 1 X 1 1 = + − 2n+1 2 n n + 2 n=0 n=1 1 1 3 = 1 + 2 1 − 2 4 7 = : 4 3 Summary • A telescoping series is a series in which all but a finite number of terms cancel. When a series has terms that are rational functions, a partial fraction decomposition can be used to determine whether the series is in fact a telescoping series. • The harmonic series, with terms 1=n, is an example of a series whose terms converge to zero, but is still divergent. • If a series converges, then its terms must converge to zero, but the converse is not necessarily true: a series whose terms converge to zero may still diverge. On the other hand, if the terms of a series do not converge to zero, then the series must diverge. • Adding or subtracting convergent series yields a convergent series, whose sum is obtained by adding or subtracting the sums of the individual series. Similarly, multiplying the terms of a convergent series by a constant multiplies its sum by the same constant. 4.