INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1

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1 Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., is clled ifite series ssocited with ifiite sequece. This series c lso be expressed i bbrevited form usig the sigm ottio, i.e., I this Chpter, we shll study bout some specil types of series which my be required i differet problem situtios. A.. Biomil Theorem for y Idex I Chpter 8, we discussed the Biomil Theorem i which the idex ws positive iteger. I this Sectio, we stte more geerl form of the theorem i which the idex is ot ecessrily whole umber. It gives us prticulr type of ifiite series, clled Biomil Series. We illustrte few pplictios, by exmples. We kow the formul ( + x) C + 0 C x C x Here, is o-egtive iteger. Observe tht if we replce idex by egtive iteger or frctio, the the combitios C k r k do ot mke y sese. We ow stte (without proof), the Biomil Theorem, givig ifiite series i which the idex is egtive or frctio d ot whole umber. ot to be republishe Theorem The formul ( ) holds wheever x <. INFINITE SERIES ( ) ( )( ) m m m m m m + x + mx + x + x

2 INFINITE SERIES 4 Remrk. Note crefully the coditio x <, i.e., < x < is ecessry whe m is egtive iteger or frctio. For exmple, if we tke x d m, we obti ( )( ) ( ) ( ) + ( )( ) or This is ot possible. Note tht there re ifiite umber of terms i the expsio of (+ x) m, whe m is egtive iteger or frctio Cosider ( ) m b m b + b + + m ( ) m b mm b + m ( ) m m m m m + m b+ b b This expsio is vlid whe < or equivletly whe b <. The geerl term i the expsio of ( + b) m is ( )( )...( + )... r We give below certi prticulr cses of Biomil Theorem, whe we ssume ot to be republishe m m r mm m m r b x <, these re left to studets s exercises:. ( + x) x + x x ( x) + x + x + x ( + x) x + x 4x ( x) +x + x + 4x +... x Exmple Expd, whe x <. r

3 44 MATHEMATICS Solutio We hve x A.. Ifiite Geometric Series x x x x From Chpter 9, Sectio 9.5, sequece,,,..., is clled G.P., if k + k r (costt) for k,,,...,. Prticulrly, if we tke, the the resultig sequece, r, r,..., r is tke s the stdrd form of G.P., where is first term d r, the commo rtio of G.P. Erlier, we hve discussed the formul to fid the sum of fiite series + r + r r which is give by S ( r ) r. I this sectio, we stte the formul to fid the sum of ifiite geometric series + r + r r +... d illustrte the sme by exmples. 4 Let us cosider the G.P.,,,... 9 Here, r. We hve ot to be republishe S Let us study the behviour of... () s becomes lrger d lrger.

4 INFINITE SERIES We observe tht s becomes lrger d lrger, becomes closer d closer to zero. Mthemticlly, we sy tht s becomes sufficietly lrge, sufficietly smll. I other words, s ot to be republishe becomes, 0. Cosequetly, we fid tht the sum of ifiitely my terms is give by S. Thus, for ifiite geometric progressio, r, r,..., if umericl vlue of commo rtio r is less th, the S ( r ) r r r r r I this cse, r 0 s sice r < d the 0 r. Therefore, S s. r Symboliclly, sum to ifiity of ifiite geometric series is deoted by S. Thus, we hve S r For exmple (i) (ii)

5 46 MATHEMATICS Exmple Fid the sum to ifiity of the G.P. ; 5 5 5,,, Solutio Here d r. Also r < Hece, the sum to ifiity is A..4 Expoetil Series Leohrd Euler (707 78), the gret Swiss mthemtici itroduced the umber e i his clculus text i 748. The umber e is useful i clculus s π i the study of the circle. Cosider the followig ifiite series of umbers ()!!! 4! The sum of the series give i () is deoted by the umber e Let us estimte the vlue of the umber e. Sice every term of the series () is positive, it is cler tht its sum is lso positive. Cosider the two sums ()! 4! 5!! d () Observe tht d! 6, which gives < 4! ot to be republishe d 4! 4, which gives < 8 4! d 4 5! 0, which gives < !

6 INFINITE SERIES 47 Therefore, by logy, we c sy tht <!, whe > We observe tht ech term i () is less th the correspodig term i (), Therefore ! 4! 5!! < Addig + + o both sides of (4), we get,!! !!! 4! 5!! < !! (4)... (5) Left hd side of (5) represets the series (). Therefore e < d lso e > d hece < e <. Remrk The expoetil series ivolvig vrible x c be expressed s x x x x x e !!!! Exmple Fid the coefficiet of x i the expsio of e x+ s series i powers of x. ot to be republishe Solutio I the expoetil series x e replcig x by (x + ), we get x x x !!!

7 48 MATHEMATICS x e + ( ) Here, the geerl term is x + Biomil Theorem s ( x+ ) ( x+ ) !!! (+ x )! C ( ) C + x + ( x) ( x).! Here, the coefficiet of x is series is C! C! ot to be republishe. This c be expded by the. Therefore, the coefficiet of x i the whole ( )!! [usig! ( ) ( )!] ( ) !!! e. Thus e is the coefficiet of x i the expsio of e x+. Altertively e x+ e. e x x ( x) ( x) e !!! Thus, the coefficiet of x i the expsio of e x+ is e. e! Exmple 4 Fid the vlue of e, rouded off to oe deciml plce. Solutio Usig the formul of expoetil series ivolvig x, we hve x x x x x e !!!!

8 INFINITE SERIES 49 Puttig x, we get e !!! 4! 5! 6! O the other hd, we hve e the sum of first seve terms < !!! 4! 5! Thus, e lies betwee 7.55 d 7.4. Therefore, the vlue of e, rouded off to oe deciml plce, is 7.4. A..5 Logrithmic Series Aother very importt series is logrithmic series which is lso i the form of ifiite series. We stte the followig result without proof d illustrte its pplictio with exmple. Theorem If x <, the x x loge ( + x) x +... The series o the right hd side of the bove is clled the logrithmic series. Note The expsio of log (+x) is vlid for x. Substitutig x i the e expsio of log e (+x), we get ot to be republishe loge

9 40 MATHEMATICS Exmple 5 If αβ, re the roots of the equtio x px+ q 0, prove tht Solutio Right hd side ( ) ( ) α +β α +β loge + px + qx α+β x x + x... α x α x β... x β αx + + βx + x... log ( +α ) + log( +β ) e x x ( + α+β +αβ ) log ( ) e x x loge ( px qx ) Here, we hve used the fcts α+β p d + + Left hd side. αβ q. We kow this from the give roots of the qudrtic equtio. We hve lso ssumed tht both α x < d β x <. ot to be republishe