CHAPTER 23 MACLAURIN S SERIES

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1 EXERCISE 97 Page 9 CHAPTER MACLAURIN S SERIES. Determine the irst our terms o the power series or sin using Maclaurin s series. Let () sin () sin () cos () cos () sin () sin () 8 cos () 8 cos 8 i( ) 6 sin i() 6 sin ( ) cos () cos ( ) 6 sin () 6 sin i ( ) 8 cos i () 8 cos 8 Maclaurin s series states: () () + () +! () +! () + i.e. () () + () + ( 8) + () + () + () + ( 8)!!! 5! 6! 7! i.e. sin Use Maclaurin s series to produce a power series or cosh as ar as the term in 6 Let () cosh () cosh () sinh () sinh () 9 cosh () 9 cosh 9 () 7 sinh () 7 sinh i( ) 8 cosh i() 8 cosh 8 59, John Bird

2 ( ) sinh () sinh ( ) 79 cosh () 79 cosh 79 Maclaurin s series states: () () + () +! () +! () i.e. () () + (9) + () + (8) + () + (79) +...!!! 5! 6! i.e. cosh Use Maclaurin's theorem to determine the irst three terms o the power series or ln( + e ). Let () ln ( + e ) () ln ( e ) + ln () e + e () e + e () ( + e ) e e ( e ) ( + e ) + e e e e () ( ) ( ) ( + e ) Maclaurin s series states: () () + () + () +! i.e. ln ( e ) + ln ! ln + + 8! () +. Determine the power series or cos t as ar as the term in t 6 Let (t) cos t () cos (t) sin t () sin (t) 6 cos t () 6 cos 6 (t) 6 sin t () 6 sin 6, John Bird

3 i () t 56 cos t i() 56 cos 56 () t sin t () sin () t 96 cos t () 96 cos 96 Maclaurin s series states: (t) () + t () + t! () + t! () + 6t 56t 96t i.e. () + 7 t t t t5 t6 + t() + ( 6) + () + (56) + () + ( 96) +...!!! 5! 6! 6 56 i.e. cos t 8t + t t Epand e (/) in a power series as ar as the term in Let () e (/) () e () e () e () 9 e 9 () e 9 () 7 e 8 7 () e Maclaurin s series states: () () + () +! () +! () !! i.e. () i.e. e (/) Deelop, as ar as the term in, the power series or sec. Let () sec () sec 6, John Bird

4 () sec tan () () ( sec )( sec ) + ( tan )( sec tan ) sec [ sec + tan ] sec [ sec + sec ] [ ] sec sec 8sec sec () 8 () ( ) sec sec tan 8sec tan 8sec tan 8sec tan () i( ) ( 8sec )( sec ) + ( tan )( sec (sec tan ) ) ( 8sec )( sec ) ( tan )( 6sec tan ) + i() Maclaurin s series states: () () + () + () +! + () + () + () + (8)!!! () +! i.e. sec + + as ar as the term in 7. Epand e cos as ar as the term in using Maclaurin s series. Let () e cos () e cos () ( e )( sin ) + ( cos)( e ) e ( cos sin ) () e ( cos sin ) () ( e )( 6sin 9cos ) ( cos sin)( e ) + () Maclaurin s series states: () () + () +! + () + ( 5)! () +! () + 6, John Bird

5 5 i.e. e cos + as ar as the term in 8. Determine the irst three terms o the series or sin by applying Maclaurin s theorem. Let () sin () sin () sin cos () sin cos () ( sin )( sin ) + (cos )( cos ) ( ) sin + cos cos sin cos () cos () sin () sin i( ) 8 cos i() 8 cos 8 ( ) 6 sin () 6 sin ( ) cos () cos Maclaurin s series states: () () + () + () +! i.e. 6 sin () +! () + () + () + ( 8) + () + () +...!!! 5! 6! + to three terms 5 9. Use Maclaurin s series to determine the epansion o ( + t) Let (t) ( + t) () 8 (t) ( + t) () 8( + t) () 8() 6 (t) ( + t) () 8( + t) () 8() (t) 96( + t)() 9( + t) () 9() 576 i( ) t 9() 8 i() 8 t Maclaurin s series states: (t) () + t () + () +! t! () + t t t 8 + t(6) + () + (576) + (8) +...!!! 6, John Bird

6 i.e. (t) ( + t) 8+ 6t+ 6t + 96t + 6t 6, John Bird

7 EXERCISE 98 Page. Ealuate.6 e sin d., correct to decimal places, using Maclaurin s series. Let () esin () e sin () ( esin )( cos ) () sin ( )( ) () ( e sin )( sin sin ) + ( cos)( e cos) e cos esin ( cos sin) () esin ( cos sin ) () ( esin)( cos sin cos sin ) + ( cos sin)( e cos) () ( esin )( cos sin cos sin ) ( cos sin )( e cos ) + ()( ) + ()() Maclaurin s series states: () () + () +! () +! () + Hence,.6 e sin d..6.6 d ( ) ( ) ( ).6. (.6) (.) + (.) + ( ) ( ) , correct to decimal places. Use Maclaurin s theorem to epand cos and hence ealuate, correct to decimal places, cos d Let () cos () cos () sin () sin () cos () cos () 8 sin () 8 sin 65, John Bird

8 i( ) 6 cos i() 6 cos 6 ( ) sin () sin ( ) 6 cos () 6 cos 6 Maclaurin s series states: () () + () +! () +! () () + ( ) + () + (6) + () + ( 6) +...!!! 5! 6! i.e () i.e. cos Hence, Hence, cos cos 5 7 d d ( ).88, correct to decimal places. Determine the alue o cosd, correct to signiicant igures, using Maclaurin s series. 6 From Problem, page 6, cos !! 6! Since then cos !! 6! 7 66, John Bird

9 Thus, cosd d () () (7) ( ).5, correct to signiicant places. Use Maclaurin s theorem to epand ln( + ) as a power series. Hence ealuate, correct to decimal places,.5 ln( + ) d. From page 7, ln( + ) Hence, 5 ln( )d d (.5) (.5) + (.5 ) (.5) + (.5) +... [ ] , correct to decimal places 67, John Bird

10 EXERCISE 99 Page. Determine: + lim lim Determine: sin sin cos cos. Determine: ln( + ) ln( + ) +. Determine: lim sin + sin + cos lim + 5. Determine: sin cos lim ( )( ) cos sin cos sin lim + sin cos lim cos + sin ( sin ) + cos () + cos , John Bird

11 6. Determine: ln t t t ln t lim lim t t t t t 7. Determine: sinh sin lim sinh sin lim cosh cos sinh + sin cosh + cos Determine: sin ln sin π sin ln sin π cos π lim lim sin sin cos sin { } π π 9. Determine: sect t tsin t sect secttan t (sec t)(sec t) + (tan t)(sec ttan t) t tsin t t tcost sin t t + t( sin t) + cost+ cost , John Bird