Further Pure 1 Revision Topic 5: Sums of Series

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Dylan Newton
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1 The OCR syllabus says that cadidates should: Further Pure Revisio Topic 5: Sums of Series Cadidates should be able to: (a) use the stadard results for Σr, Σr, Σr to fid related sums; (b) use the method of differeces to obtai the sum of a fiite series; (c) recogise, by direct cosideratio of the sum to terms, whe a series is coverget, ad fid the sum to ifiity i such cases Sectio : Usig stadard formulae for r, r, r You eed to lear the formula: r r ( ) The followig formulae are give i the formula book: r r ( )( ) r ( ) r 4 Note: (this should be obvious to you if it is t the you eed to remember it!) r The above formulae ca be used to fid the sum of series Worked examiatio questio (Edexcel Jue 004) (a) Show that ( r )( r 5) = (+ 7)( + 7) (4) r (b) Hece calculate the value of Solutio: a) wwwtheallpaperscom 40 r0 ( r )( r 5) () ( r )( r 5) ( r r 5) (expadig out brackets) r r We ext split up the summatio ito several separate sums: ( r )( r 5) r r 5 r r r r

2 So, usig the results from above: ( r )( r 5) ( )( ) ( ) 5 = ( )( ) ( ) 5 r We ca try factorisig this by takig out as a factor: r ( r )( r 5) ( )( ) 8( ) 0 Expadig out the cotets of the square brackets gives: r ( r )( r 5) The cotets of the squared brackets ca ow be factorised We get the result: b) So: r ( r )( r 5) = (+ 7)( + 7) ( r )( r 5) ( r )( r 5) ( r )( r 5) r0 r r 40 r0 Therefore, 40 r0 ( r)( r 5) 40(47)(87) 9 5 ( r)( r 5) Examiatio questio: Edexcel Jue 00 Prove that r ( r ) ( )( + 5) wwwtheallpaperscom

3 Examiatio questio (AQA Jue 005) a) Use stadard formulae to show that r ( r ) ( )( ) r b) Use the result from part (a) to fid the value of r ( r) r4 Examiatio questio (AQA 00) a) Fid the sum of the itegers from to 00 iclusive b) Evaluate: 00 ( r r) r wwwtheallpaperscom

4 Examiatio questio (OCR May 004) (i) Use the formula for r to show that (ii) (iii) Show also that ( ) ( ) 4 ( ) ( ) Hece fid the sum of the series simplifyig your aswer 4 ( ) ( ), wwwtheallpaperscom

5 Sectio : Method of differeces Some series ca be summed usig a differece method Worked examiatio questio (AQA 00) a) Show that r r ( r ) r ( r ) b) Hece fid the sum of the first terms of the series Solutio: a) Writig with a commo deomiator, we get Therefore, b) We wish to fid ( r) r r ( r ) r ( r ) r ( r ) r r r r r ( r ) r ( r ) r ( r ) r r r ( r) Usig the result from (a), this is equivalet to fidig r r ( r) Substitutig r =,,, ito this summatio gives: 4 ( ) Therefore, = ( ) r r ( r) ( ) Note: The series give i the above questio is coverget As the value of icreases (ie as the umber of terms gets bigger ad bigger), the sum of the series approaches (sice the value of will ted to zero) ( ) We write = r r ( r) wwwtheallpaperscom

6 Worked examiatio questio (Edexcel Ja 004) (a) Show that (r + ) (r ) Ar + B, where A ad B are costats to be foud (b) Prove by the method of differeces that r = ( + )( + ), > Solutio a) O removig the brackets we fid that r (r + ) (r ) = r r r r r r So, A = ad B = b) So, (r ) ( r ) ( r ) r r = r Workig out the right had side of this equatio (by substitutig r =,,, etc) gives: ie (r ) ( 0 ) ( ) ( 4 ) ( 5 r st term d term rd term 4th term r ( 4 5th term (r ) ( ) But the left-had side of this equatio is: Therefore: So: ie ) ( ( ) r r r r th term (r ) r r r = r ( ) ) ) (( ) ( ) th term r = ( ) ( ) r r = r So we get the result: r ( ) ( )( ) r = ( + )( + ) Examiatio questio (Adapted from Edexcel Jue 005) (a) Show that = 4r r r (b) Hece, or otherwise, prove that = 4r r ) wwwtheallpaperscom

7 (c) Hece fid the exact value of 0 r 4r Examiatio questio (adapted from Edexcel Ja 005) (a) Show that r( r ) r ( r ) (b) Hece prove, by the method of differeces, that 4 ( 5) = r( r ( )( ) 00 (c) Fid the value of 4 r 50 r( r ) r ), to 4 decimal places wwwtheallpaperscom

8 Examiatio questio (AQA Ja 005) a) Show that r ( r ) ( r ) r 4r b) Use the method of differeces to fid the value of 00 r r50 Examiatio questio (AQA Jue 004) a) Show that r( r ) ( r )( r ) r( r )( r ) b) Hece fid the sum of the series givig your aswer as a ratioal umber, wwwtheallpaperscom

9 Examiatio Questio (OCR Jauary 005) 4 You are give that f() r ( r)( r) (i) Show that f() r r r (ii) Hece fid f() r (You eed ot express your aswer as a sigle fractio) r (iii) Show that the series i part (ii) is coverget, ad state its sum to ifiity wwwtheallpaperscom

10 Worked examiatio questio (OCR May 005) a) Show that b) (i) Give that (r ) ( ) r f() r rr ( ), show that f ( r ) f ( r) r( r )( r ) (ii) Hece fid r r( r )( r ) Solutio: a) (r ) (9r r ) 9 r r r r r r r Usig the stadard formulae give i sectio of these revisio otes, we see that r (r ) 9 ( )( ) ( ) = ( )( ) ( ) Takig out ½ as a factor, we get: r (r ) ( )( ) ( ) 9 b) If f() r rr ( ), the f( r) ( r)( r) So, f ( r ) f ( r) = - ( r)( r) rr ( ) r r Therefore, f ( r ) f ( r) = r( r )( r ) r( r )( r ) r( r )( r ) (ii) = r r( r )( r ) Therefore, r r r r ( f ( r ) f ( r)) r( r )( r ) r r = ( f () f()) ( f() f () ) ( f (4) f () ) ( f ( ) f ( ) ) ( )( ) ie r r( r )( r ) But, = f ( ) f () wwwtheallpaperscom

11 f () ad f( ) ( )( ) Therefore = f () f ( ) r r( r )( r ) ( )( ) 4 ( )( ) Examiatio questio (AQA Jue 00) a) Show that r r! ( r )! ( r )! b) Hece fid r r ( r )! Hit: I part (a), use the result that (r + )! = (r + )r! wwwtheallpaperscom

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physicsandmathstutor.com physicsadmathstutor.com 9. (a) A geometric series has first term a ad commo ratio r. Prove that the sum of the first terms of the series is a(1 r ). 1 r (4) Mr. Kig will be paid a salary of 35 000 i the