Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad for the series f ( a) f( a) f( a)... f( b), where ka a ad b are itegers such that a b. e.g. r r ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 8) 6 5 (i 7) ( 7) ( 7) ( 7) ( 5 7) i Arithmetic Sequeces 5 68 A arithmetic sequece is oe with a commo differece (i.e. where the et term is obtaied by addig or subtractig a costat umber to the previous term, as i the recurrece relatio u u d ), such as,, 5, 7, 9, or, 5,, 5,. k (k ) () ( ) ( ) 57 5 Notatio: A geeral arithmetic sequece has first term a ad commo differece d. The th term is deoted by u. If d > the the sequece is icreasig, if d < the it is decreasig. The fial formula (for the partial sum) is give o the formula sheet i the eam ad i uit assessmets, ad the other formulae are ot give. for evaluatio oly Formulae For a arithmetic sequece, d ca be obtaied by subtractig two cosecutive terms. the th term is give by the formula u a ( ) d Ad the partial sum of the first terms is give by the formula S a( ) d Eample For the arithmetic sequece 7,,, 6, fid (a) the 7 th term, ad (b) a formula for the th term.
Solutio By ispectio the first term a = 7, ad the commo differece d =. Usig the formula u a ( ) d gives: (a) u7 a 6d (b) u a( ) d 7 6 7( ) 85 7 Sometimes you are ot told the first term, or eve two cosecutive terms, but as log as you are give ay two terms, you ca work these out. Eample I a arithmetic sequece, u = 6 ad u 5 = 5. Fid: (a) The first term ad the commo differece (b) The 5 th term of the sequece. (c) The sum of the first 6 terms Solutio (a) Usig the formula u a ( ) d, we have u a d, ad u5 a d meaig that u5 u d (hopefully this result should make sese whe you thik about it) (b) (c) Thus d = 5 6 =, so d = 5. u a d 6 a d 6 a a 7 i.e. the first term is 7 ad the commo differece is 5. u a d 5 7 ( 5) S a ( ) d S6 6 7 59 ( 5) ( 95) 65 Eample Fid the sum of the first 8 terms of the arithmetic series 9 5... Solutio O ispectio, we see that a ad d 6. gives us: Usig these values i the formula S a ( ) d for evaluatio oly 97 The sum of the first 8 terms is 97. 8 S8 () 7(6) 9(6 )
Eample Fid the sum of the arithmetic series 9 6... 85. Solutio By ispectio, a = ad d = 7. We must fid. We kow that u 85, therefore Now usig the formula S a ( ) d The sum of the series is 59. Geometric Sequeces S : () 9(7) 97 59 A geometric sequece is oe with a commo ratio (i.e. where the et term is obtaied by multiplyig the previous term by a costat, as i the recurrece relatio u ru), such as,,, 8, 6, or 8, 7, 9,,,,. Notatio: A geometric sequece has first term a ad commo ratio r. The th term is deoted u. If r the the sequece is icreasig, if r the the sequece is decreasig. If r > the all terms are either all positive or all egative (depedig o the first term). If r <, the the terms alterate from positive to egative e.g.,,, 8, 6, The fial formula (for the partial sum) is give o the formula sheet i the eam ad i uit assessmets, ad the other formulae are ot give. Formulae For a geometric sequece, r ca be obtaied by dividig two cosecutive terms. The th term is give by the formula u ar Ad the partial sum of the first terms is give by the formula a( r ) ar ( ) S, ( or S ) r r (either versio of this fial formula ca be used, though it makes for easier calculatios to use the first oe whe r < ad the secod oe whe r > ) Eample Fid the th term of the geometric sequece, 8,, 8,... Solutio O ispectio, a = ad a ( ) d 85 ( )7 85 7 7 85 7 585, i.e. there are terms i the series for evaluatio oly 8 r. u ar Thus: u ar 8 88 68
Eample A geometric sequece of positive terms has third term 8 ad seveth term 58. Fid the fifth term of this sequece. Solutio u ar ar 8 u...() 6 u7 ar 58...() 6 ar 58 ar 8 () () r 8 r But we were told i the questio that all terms are positive, so r, hece r. Substitutig ito (): ar 8 a 8 9a 8 a u ar u5 ar 6 Eample Fid the sum of the first 9 terms of the geometric series 8 6... Solutio By ispectio, a = ad r =. Thus: Eample Solutio Evaluate ( 9) k, givig your aswer correct to decimal places. k k ( 9) k 9 ( 9) ( 9)... ( 9) This is a geometric series of terms with a 9 ad r 9. a( r ) S r 9 (9) S 9 9 (9) 7 96 ( d.p.) S S 9 ar ( ) r 9 ( ) 9 ( ) for evaluatio oly
Eample 5 Evaluate the sum of the geometric series... 6 5. Solutio By ispectio, a = ad r = 5. However we must fid. We kow that u = 65 ar 65 5 65 5 565 This equatio ca be solved by takig atural logarithms of both sides of the equatio: l(5 ) l565 ( ) l 5 l565 l565 l 5 6 7, i.e. there are 7 terms i the series 7 ar ( ) (5 ) Usig S gives us S7 r 5 7 (5 ) 78 Ifiite Geometric Series If r, the successive terms will get icreasigly smaller. This meas that the sum of the first terms approaches as a limit as. We say that this geometric series has a sum to ifiity, deoted S. The followig formulae is ot give o the formula sheet i the eam ad i uit assessmets. Formulae a The sum to ifiity of a geometric series whe r is give by S r Eample Fid the value of the sum to ifiity of the geometric series 6 7 5..., eplaiig why it is eists. Solutio 7 a 6 ad r 6, so the geometric series has a sum to ifiity. for evaluatio oly a S r 6 6 i.e. the sum to ifiity is
Eample For the series... 8 (a) State the first term ad the commo ratio. (b) State the rage of values of for which a sum to ifiity eists. (c) Fid a epressio for the sum to ifiity whe is i this rage of values. Solutio (a) a = ad r (b) For a sum to ifiity to eist, the commo ratio must be betwee ad. (c) Whe takes these values: a S r Maclauri Series A power series, is where we epress a fuctio as a ifiite sum of the form k ao a a a... a... (or a ). k k All fuctios ca also be represeted as a Power series (also called a Maclauri series) i this way though it is importat to realise that they oly coverge (i.e. they oly work) for specific rages of though for some fuctios this rage is all real umbers. for evaluatio oly The followig formula is give o the formula sheet i the eam ad i uit assessmets. Formula: the Maclauri series for f ( ) is give by: ( ) ( k) f () f () f () f () f () f ( ) f()......!!!! k! Eample Fid the Maclauri series for the fuctio f ( ) ta up to ad icludig the term i ³. (ote this series oly coverges whe ). Solutio f( ) ta, so f() ta f '( ) (as leart i the first uit), so f () k k
f ( ) (usig chai rule), so f () ( ) ( ) ( ( ( ) ) f (usig quotiet rule ad simplifyig), so f () ) ( ) ( ) The usig the Maclauri series formula, f () f () f () f( ) f()...!!!...!!!... 6... It is also very useful to kow a few stadard power series, as kowig these ad beig able to just write them dow ca make some questios quicker. The followig formulae are ot give o the formula sheet i the eam ad i uit assessmets. Formulae: stadard Maclauri series e!!! coverges for all. 5 7 si! 5! 7! coverges for all. 6 cos!! 6! coverges for all. 5 7 ta... 5 7 coverges for. coverges for. coverges for. l( ) coverges for. l( ) coverges for. Eample Fid the Maclauri series for the fuctio ( ) f e up to ad icludig the term i. Solutio There are two methods: Method (a) from first priciples Method (b) memorise the series for e ad use that. for evaluatio oly
Method (a) f e f e ( ), so () f e f e ( ), so () f e f e ( ), so ( ) f e f e ( ) 8, so () 8 8 f ( ) 6 e, so f () 6e 6 The usig the Maclauri series formula, f () f () f () f( ) f()...!!! 8 6...!!!! 8 6... 6... Method (b) We kow that if f ( ) e, the f( )!!! For e, we ca use the series for f ( ) - i.e. we replace every occurrece of with. e f ( ) ( ) ( ) ( ) ( )!!! 8 6 6 Which is the same result as i method (a). Eample Obtai the Maclauri series for the fuctio f ( ) e si up to ad icludig the term i ³. Hece write dow the Maclauri series for the fuctio ( ) g e si. for evaluatio oly Solutio Part () Method (a) from first priciples All derivatives are obtaied usig the product rule, but the workig has bee left out, but oly to keep this eample shorter. You would be epected to show full workig i a eam situatio though. f e f e ( ) si, so () si f e f e ( ) (si cos ), so () (si cos ) f e f e ( ) cos, so () cos f e f e ( ) (cos si ), so () (cos si ) f ( ) e si, so f () e si
The usig the Maclauri series formula, f () f () f () f( ) f()...!!!!!!! 6 Part () Method (b) use kow series for......... e ad si. We kow that 5 7 e ad si.!!!! 5! 7! This meas that: 5 7 e si!!!! 5! 7! We ca multiply these brackets together by: multiplyig everythig i the secod bracket by stoppig whe we first get a power larger tha (because the questio asked up to stop at ). the multiplyig everythig i the secod bracket by stoppig whe we first get a power larger tha. i.e. the multiplyig everythig i the secod bracket by stoppig whe we! first get a power larger tha etc. 5 7 e si!!!! 5! 7!!!!! ( )... (all other terms give a power higher tha )!! 6...!!... for evaluatio oly... (which is the same aswer obtaied usig method (a)) Part () We kow that e si... Replacig with i this power series gives: e si ( ) ( )... 8...