Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Transcription

1 Jim Lmbers MAT 169 Fll Semester Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of the terms of n infinite sequence. Specificlly, let { n } be sequence. Then we cn define series to be the sum of the terms of { n }, n +. We refer to the terms of { n } s the terms of the series. Writing series in this mnner is cumbersome, so we insted use sigm nottion to write this series s n. The expression below the upper cse Greek letter sigm indictes wht nme is given to the index (in this cse, n), nd its initil vlue (in this cse, 1). The expression bove the sigm indictes the finl vlue of the index, or for n infinite sum. Note tht this mens sigm nottion cn be used to represent finite sums s well.in other words, if we wrote, for exmple, 10 bove the sigm insted of, then we would be specifying tht only the first 10 terms of { n } should be dded. Tht is, 10 n = For either finite or infinite sum, the expression to the right of the sigm is wht is to be dded, for ech vlue of the index. This mens tht if n is the nme ssigned to the index, then every occurrence of n within the summed expression is to be replced with ech vlue of the index, nd the resulting terms re summed. Exmple The finite sum 4 n + 1 1

2 is evluted s follows: 4 n + 1 = = = We now need to define wht it mens to compute the sum of infinitely mny terms. For this concept, sequences ply their most importnt role. Given n infinite sequence { n } of terms to be summed, we define sequence of prtil sums, denoted by {s n }, s follows: s 1 = 1 s 2 = s = s 3 = s = s n = s n 1 + n = n 1 + n. Then, we cn view the series s sequence of prtil sums. This leds to the following definitions. A series is sid to be convergent if its sequence of prtil sums, {s n }, is convergent. The limit of {s n}, if it exists, is clled the sum of the series. If the sequence of prtil sums is divergent, then we sy tht the series is divergent. Exmple Consider the series The sequence of prtil sums is s 0 = 1 n 1. s 1 = s

3 = 3 2 s 2 = s = 7 4 s 3 = s = s n = 2n+1 1 = 2n (2 2 n ) = 2 1. The prtil sums converge to 2, so we sy tht 2 is the sum of the series. Why Do We Need Series? Series re pplied throughout mthemtics, s well s physics, computer science, nd vrious brnches of engineering. They re prticulrly useful for describing functions or solutions of equtions using sums of simple functions such s polynomils or bsic trigonometric functions. They re lso useful for nlyzing the performnce of numericl methods for solving equtions. Geometric Series The series in the preceding exmple is geometric series. The generl form of geometric series is r n, where r is clled the common rtio of the series, becuse ech term in the series, for n 1, is obtined by multiplying the previous term by r. This type of series rises in vriety of pplictions, such s the nlysis of numericl methods for solving liner or differentil equtions. We now try to determine whether geometric series converges, nd if so, compute its limit. To do this, we exmine the sequence of prtil sums. We hve s n+1 = + r + r r n + r n+1, rs n = r + r r n + r n+1, 3

4 which yields the reltion s n+1 = + rs n. We lso hve s n+1 = s n + r n, by the definition of prtil sum. Equting these, nd rerrnging, yields (1 r n ) = s n (1 r), which, for r = 1, leds to closed-form representtion of the nth prtil sum, s n = 1 rn 1 r. We cn now determine convergence of the geometric series: If r = 1, the nth prtil sum is s n = n, nd therefore the series diverges. If r = 1, the numertor in s n oscilltes between 0 nd 2, so the series gin diverges. If r > 1, then r n diverges, so due to its presence in the numertor of s n, the series diverges. Finlly, if r < 1, then r n 0, nd the series converges to lim s n = n We now consider severl exmples of geometric series. Exmple The series e n 10 n 1 1 r. cn be viewed s geometric series, but we must be creful, becuse geometric series uses n initil index of zero, while the initil index for this series is 1. Therefore, we must first rewrite the series to use n initil index of zero before determining the vlues of nd r. Since we wish to subtrct 1 from the initil index, we must compenste by replcing n by n + 1 throughout the expression to be summed. This yields the equivlent series e n+1 10 n = e e n 10 n. This is geometric series with = e nd r = e/10. Since e , we hve r < 1, nd therefore the series converges to 1 r = e 1 e 10 = 10e 10 e. 4

5 Exmple Consider the series 2 2n 3 n. Using the lws of exponents, we rewrite this series s (2 2 ) n 3 n = It follows tht this is geometric series with = 1 nd r = 3/4, which converges to Exmple The series 1 r = (x 2) n is n exmple of power series, since the terms re constnts times powers of (x 2). We will see much more of power series, s they re very useful for pproximting functions in wy tht is prcticl for implementtion on clcultor or computer. This prticulr power series is lso geometic series with = 1 nd r = (x 2)/2. Therefore, it converges if (x 2)/2 < 1, which is true if 2 < x 2 < 2, or 0 < x < 4. Exmple Geometric series cn be used to convert repeting decimls into frctions. Consider the repeting deciml This cn be written s the infinite series 3 n 4 n. = n, which is geometric series with = /10 6 nd r = 10 6, where the 6 rises due to the fct tht the sequence of repeting digits hs 6 terms: 1, 4, 2, 8, 5 nd 7. This is convergent geometric series, nd its limit is 1 r = ( ) = = =

6 Summry An infinite series, or simply series, is the sum of the terms of sequence. Sigm nottion provides concise wy of describing series, using only its initil index, finl index (or ), nd definition of ech term. The prtil sum of series is the sum of its first n terms, for ny vlue of the index n. A series converges if the sequence of its prtil sums converges; otherwise, it diverges. A geometric series is ny series whose terms re of the form r n, for n 0. The number r is clled the common rtio. If r < 1, the series converges to ; otherwise, it diverges. 1 r 6