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1 Balliteer Istitute wwwleavigcertsolutioscom Leavig Cert Higher Maths Sequeces ad Series A sequece is a array of elemets seperated by commas E,,7,0,, The elemets are called the terms of the sequece ie is the first term, is the secod term etc Geeral Term ( ) : This is a epressio i which describes every term of a particular sequece, the geeral term of the sequece,,7,0 is - A sequece ca be describes i several ways () A sequece ca be described usig just the A sequece + fid the first three terms 7, 5, Fid + (+) + +7 () A sequece ca be described recursively (a differece formula coectig up cosecative terms of a sequece) 5 Eample If 5, + Fid 5 5, ( 5 ) +, ( ) + 6 The Syllabus requires you to be able to fid the Sum of the first terms of all of the followig types of Sequeces : () Arithmetic a + ( ) d () Geometric ar () AP/GP a () Telescopic type ( )( + ) Lim Lim You will also be required to fid the followig limits, r r + Series A series is the sum of a sequece, we use the symbol which meas "the sum of " to describe a series this is the series whose is from to ie the sum of the first terms of the series S this is called the S of the Series S ad the sum to ifiity of a series is α Lim S You will be required to fid the S ad the sum to ifiity of all of the α sequeces described below The syllabus states that you ca oly be asked to cosider Series whose S ca be foud ad you will oly be asked to fid limits for sequeces where is give eplicitly Must Kow Arithmetic Series a + (-)d, S / {a+ (-)d} a( r ) a Geometric Series ar S r < S ( r) r 6 ( + ) ( + )( + ) { ( + )} Telescopic Series a b + ( + ) + (Partial fractios)

2 Arithmetic Series : Key poits : I this type of Series a costat is added to each term to give the et term This costat is called the commo differece (symbol d), i all Arithmetic Series + + If the first term is( a ) the a Arithmetic Series ca be writte as a + a+d +a+d+a+d + The Geeral term is a + ( ) d,the Sum of the first terms is S { a + ( ) d} Eample : Give the Arithmetic Series , fid a,d,,s, a, ( 5 - ) d, + (-), - S /{() + (-)} /{ - } Eample : the first three terms of a Arithmetic Sequece are 6,-9, fid , Eample :I a Arithmetic Sequece The sum of three cosecative terms is -9 ad there product is 8 Method write the three terms as a-d, a, a+d a d + a+ a+ d 9 a 9 a ( a d)( a)( a+ d) a( a d ) 8 ( 9 d ) 8This gives 9 d 6 d 5 d ± 5 us the followig sets of three terms -8-,-,8 Eample : I a Arithmetic Sequece a d, d d 8, a 6 a+ 5d 6 Eample 5 : S of a Arithmetic Sequece is + Fid,, d, S ( ) + ( ) 5, S ( ) + ( ) 6 + S S d 6 5+ ( ) 6 6 Geometric Series : I this type of series each termis multiplied by a costat to give the et term This costat is called ( r ) the commo ratio I all geometric series r ( ) cosecative terms of a geometric series ca be writte as a + ar + ar + or a/r + a + ar + The geeral term is ar a r a r The Sum of the first terms iss ( ) r < S ( ) r > r r a The sum to ifiity of a Geometric Series eists oly if r <, S r Eample :Fid the Sum of the first terms of the sequece ( ),,,, a, r, S Eample :Fid the Sum to ifiity of the Series + + > 0 + ( ), a, r S + ( + ) + + Eample : The Sum of three cosecative terms of a Geometric Series is 70 their product is 6000/7 Fid two possible such series

3 Write the terms as a/r,a, ar, a r aar 6000 a 6000 a r r+ 0r 70r 0r 70r+ 0 0 r 7r+ 0 r ( r )( r ) 0 r, r, Series, 0 /, 0 /, 60 / : or, 60 /, 0 /, 0 / Eample : Show that the of the Sequece 5,55,55,5555, ca be writte as the S of aother Sequece If we rewrite the sequece as follows 5, 5 + 5(0), 5+5(0)+5(0), 5,5(0)+5(0) +5(0) + This sequece is Geometric where a 5, r 0 ad each term of the origial sequece is the sum of the correspodig umber of terms of the ew sequece S S 50 5 S ( ), ( ) Arithmetico-Geometric Series "APGP" I this type of Series each term of a Arithmetic series is multiplied by the correspodig term of a Geometric series E ; the Series ( ) is a APGP The Ap is ,(-) The GP is, To fid S of a APGP : () Write out the Series icludig ie S () Multiply both sides by the commo ratio of the GP () Take lie () from lie () () You will be left with a "first term" a Gp ad a "ed Bit " (5)Fid the Sum of the Gp the divide your result by (-r ) Eample : Fid S of the series ( ) ( ) S ( ) 5 ( ) S ( ) ( ) ( ) S( ) + { } ( ) ( ) ( ) S( ) + ( ) ( ) ( ) ( 5) S + ( ) The Gp has a, r ad it has (-) terms We fid S(-) usig the formula for S is ot icluded i the GP ad the ed bit is (- ) - Telescopic Series (Series ivolvig fractios) : These are Series of the form + + ( + ) ( + ) ( )( + ) 5 57 ( )( + ) To fid S of this type of Series () Write as a pair of partial fractios ()Rewrite the series usig the "ew " () Simplify to fid S Fid the sum of the followig Series

4 A B A B ( ), ( ) + + ( + ) ( + ) A ( + ) + B, 0 A, B ( + ) + S Fid The Sum of the Series : ( )( ), A B + + ( )( + ) ( ) ( + ) A( + ) + B( ), A, B, { } ( ) ( + ) { } { } 5 { } 5 7 { ) ( ) ( + ) S { ( + ) } The,, these Series are () the Sum of the first Natural umbers () The sum of the squares of the first Natural umbers,() the sum of the cubes of the first atural umers a AP, S ( +) Fid a epressio for S for the series whose is There are several ways to do this () is to use the idetity - (-) - + The let,,, etc Add up the results ad Simplify to fid S ( ) + 0 ( ) ( ) + ( ) ( ) + ( ) ( ) + ( ) + + ( ) + + ( + ) + ( 6 + )( + )

5 Method() Show by Iductio That ( 6 + )( + ) ()Show true for ie Show S /6{() +)(+)}6/6 true ()Assume true for k ie that Sk k/6{(k+)(k+)} ()Prove true for (k+) ie S k+ (k+)/6{[(k+)+](k+) }(k+)/6{(k+)(k+)} Proof k k k k Sk S k k k k k k k k k k + + k k ( )( ( ) { ( ) ( )} 7 6 ( + )( + ) this is S k+ Therefore the statemet is true for k+ so the statemet is true for ad true for k+ therefore it's true for all N Show by Iductio that ( ) ( +) We wat to show that S for the Series {/(+)} () Show true for {/(+ )} true k () Prove true for k ie S k ( k + ) () Prove true for (k+) ie ( k + )( k + ) S k + We use the same techique as above ie write S k + S k + k + This gives k k ( k + ) + ( k + ) ( k + ) { + ( k + )} ( k + ) Therefore the statemet is true for k+ sig the type Series Eample ( + ) Lim Lim Lim + k + k + ( k + )( k + ) Lim Eample : Fid a epressio for ( + )( + ) ( )( ) ( ) ( + )( + ) Eample : I a Geometric Series for a, ar, ar ar ( ) Fid a epressio ( )( ) 5 a ar ar ar ar ar ar a r { } a { r } More Series :

6 If you caot remember the methods for fidig Epressios for,, Aother way to fid a epressio for the above is to use a method called Newto's Differece formula The method ivolves subtractig cosecative terms util you get a lie of zero's The usig the Biomial coeffeciets to fid the sum of the series Fid a epressio for c c ( ) ( )( ) + + c ( ) + +!! ( + )( + ) { } { + + } 6 The method works for all the Sigma series Series The Optio : Covergece A series is said to coverge if the S k Lim ie the limit of the Sum eists The problem with this defiitio is that it depeds o fidig the S Aother defiitio which requires oly the is called the Ratio Test This states that a Series will coverge if Lim + < icoclusive ad the Series will diverge iflim Eample : eamie for covergece the series + +, + Lim u u + + Lim + +, Lim >, If The Series coverges if < The series diverges if > Eample Test the Series for covergece! Lim u Lim !, +! :! X ( + )! The Series coverges Lim + the test is + + Lim + Lim + 0 Special type of Series the MacLauri Series

7 This is a method which eables us to write fuctios such as e, Si, Loge as a Series The theory goes as follows If you are Give a polyomial f() such that f( ) a + a + a + a + a a f( 0) a 0 f'( ) a + a + a + a f'( 0) a f ''( 0) f''( ) a + 6a+ a f''( 0) a a f ( 0) f ( 0) f ( ) 6a + a f ( 0) 6a a 6! f ''( 0) f ( 0) f ( ) f( 0) + f'( 0) + +!! 0 This eables us to write the polyomial as a power series : Eample write e as a power series 0 f( ) e f( 0) e, f'( ) e f'( 0), f''( ) e f''( 0) + e !!! + + Eample Write f() Si as a power series f(0) Si (0) 0, f'() Cos,f'(0) Cos (0), f''() -Si,f''(0) -Si (0) 0, f'''() - Cos, f'''(0) - We ca see that a patter is begiig to emerge Si Si +!! 5!! 5! 7! Eample : Write the first four terms of the Maclauri series for f( ) + f( ) + f( 0), f'( ) / { + } f'( 0), f''( ) / { + } f''( 0) 5 f ( ) / 8{ + } f ( 0) / ! 8 6 Sice the epasio coverges for - < <, use the epasio to evaluate 0 correct to place of decimals We must first write 0 i the form + as follows we ow replace the '' i the epasio by /9 This gives 9 {+/(/9) - /8(/8) +/6{/'79) } ( } 69 to oe place of decimals Maclaure Series for,, ta The problem with fidig a Maclauri Series for this type of fuctio is that after a very short time the differetiatio becomes very complicated : There are several ways aroud this

8 Eample : f () f () 0,'() f ( + ) f '() 0,''() f ( + ) f''() 0, f () 6 ( + ) f () this gives us the MacLaure Series for !!! We ca use this series to fid the Maclauri series for log e ( + ) as follows d + d 5 6 log e ( + ) From above the Maclauri Series for + + so a quick way to + fid the Maclauri Series for ta is to do the followig ta + + d We ca use the series for Ta-X to fid a value for Π X+ Y ta X+ ta Y ta { } We use this ad the fact that XY Π ta + ta ta ta ta 5 So replace i the series above first by 6 / the by / add the results this gives Π 5 ta + : ow multiply your aswer by ta Π Π this is oly a approimatio for Π a better result could be foud usig more terms of the series

e to approximate (using 4

e to approximate (using 4 Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial