8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to calculate accurate umerical estimates of the values of these fuctios Most of the ifiite series that we ecouter i practice are kow as power series I calculus the primary problem is decidig whether a give series coverges or diverges I practice, however, the more crucial problem may actually be summig the series If a coverget series coverges too slowly, the series may be worthless for computatioal purposes O the other had, the first few terms of a diverget asymptotic series i some cases may give excellet results Improper itegrals are used i much the same fashio as ifiite series, ad, i fact, their basic theory closely parallels that of ifiite series They are ofte classified as belogig to improper itegrals of the first kid or improper itegrals of the secod kid Field Guide to Special Fuctios
Ifiite Series ad Improper Itegrals Series of Costats 9 If to each positive iteger we ca associate a umber S, the the ordered arragemet S1, S2,, S, is called a ifiite sequece If lim S S (fiite), the ifiite sequece is said to coverge to S; otherwise it diverges Ifiite series: u1u2 uk uk k1 If S S as, where S is the sequece defied by S u1u2 u, the the series coverges to S; otherwise it diverges The letter k used i the summatio is called a idex; ay other letter like ca also be used Geometric series: The special series 2 k 1rr r r k 0 is a example series where r is the commo ratio It has bee show that 1 k 1 r k 1 r, r 1 ad r, r 1 1r 1r k0 k0 Harmoic series: Although it diverges, the particular 1 series is quite importat i theoretical work 1 Alteratig series: Series of the form ( 1) u, u 0, 1,2,3, 0 Alteratig harmoic series: The special alteratig 1 ( 1) series defied by differs from the harmoic 1 series above i that this series coverges Field Guide to Special Fuctios k
10 Special Fuctios Operatios with Series The series u is said to coverge absolutely if the related series u coverges If u coverges but the related series u diverges, the u is said to coverge coditioally I some applicatios the eed arises to combie series by operatios like additio ad subtractio, ad multiplicatio The followig rules of algebra apply: 1 The sum of a absolutely coverget series is idepedet of the order i which the terms are summed 2 Two absolutely coverget series may be added termwise ad the resultig series will coverge absolutely 3 Two absolutely coverget series may by multiplied ad the resultig series will also coverge absolutely Formig the product of two series leads to double ifiite series of the form am bk Am, k A k, k m0 k0 m0k0 0k0 Note that the result o the right is a sigle ifiite series of fiite sums rather tha a product of two ifiite series Aother product of series leads to the idetity [ /2] Amk, Akk, A2 kk, m0k0 0k0 0 k0, where / 2, eve 2 ( 1)/2, odd Field Guide to Special Fuctios
Ifiite Series ad Improper Itegrals Factorials ad Biomial Coefficiets 11 A product of cosecutive positive itegers from 1 to is called factorial ad deoted by! 123, 1,2,3, The fudametal properties of the factorial are 0! 1! ( 1)!, 1,2,3, The biomial formula, or fiite biomial series, is defied by k k ab a b k The special symbol k 0!, 0,1,2, ; k 0,1,2,,, k k! k! is called the biomial coefficiet Basic idetities for the biomial coefficiet iclude the followig: ( a) 1 0 ( b) 1 1 () c k k ( d) r r rr ( 1) ( rk1) 1,, k 1,2,3, 0 k k! () e r 1 (1) k rk k k Field Guide to Special Fuctios
12 Special Fuctios Factorials ad Biomial Coefficiets Example Use the defiitio of the biomial coefficiet ad properties of factorials to show that 1/2 ( 1) (2 )! 2 2, 0,1,2, 2 (!) Solutio: From defiitio, we obtai 1/2 ( 1/2)( 3/2) (1/2 )! ( 1) 135 (2 1) 2! Multiplyig the umerator ad deomiatio o the right by the product 246 (2) ad rearragig the umerator terms, we fid 1/2 (1)123456 (21)(2) 2 246 (2 )!, ( 1) (2 )! 2 2 123! which ca be writte as 1/2 ( 1) (2 )! 2 2, 0,1,2, 2 (!) Field Guide to Special Fuctios