MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES

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MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio Eplai the Sigma otatio. Eplai the biomial coefficiet ad elate it to Pascal's tiagle. Eplai covegece ad divegece i a mathematical pogessio. Eplai ad detemie the limitig vales of mathematical pogessios. Eplai ad se the biomial theoem. Eplai ad se Maclai's seies. Yo shold jdge yo pogess by completig the self assessmet eecises. D.J.D www.feestdy.co.k

PROGRESSIONS AND SERIES ARITHMETIC Coside the seqece of mbes,5,7,9,,. This is a pogessio that stats at ad is iceased by each time so we say the commo diffeece is d ad the statig vale is a. The sm of tems is called a ARITHMETIC SERIES. To fid this we wold have to add them all p. The sm S wold be: S a ( a d) a d d a d d d... a d d... ( ) ( ) ( ) ( a d ) ( a d ) ( a d)...{ a ( ) d} d S a ( a d) It ca be show that the aswe is () (aveage vale). The aveage vale is fod by addig the fist ad last tem ad dividig by so we ca wite: { a ( ) d} a ( ) a d S S a ( ) d S S { a ( ) d )} This is the geeally accepted fomla fo evalatig a pogessio. WORKED EXAMPLE No. Show that the fomla deived above gives the coect aswe fo the sm of the pogessio: 579 Simply addig them p gives S 5 Fo this pogessio 5, d ad a S a ) d Usig the fomla gives { ( )} { ( 5 ) } {( 8) } 5 5 5 WORKED EXAMPLE No. Evalate the fist tems of the pogessio 8. a (the fist vale) d (the diffeece betwee each mbe) Usig the fomla gives S { a ( ) d )} S {( )( ) ( ) } { 8 ( 9)( ) } S 8 D.J.D www.feestdy.co.k

WORKED EXAMPLE No. A mafacte of steel bas oly sells them i batches of m ad calclates the chage to cstomes sig the followig system. - m - m - m ad so o with a maimm ode of m Wite ot the aithmetic pogessio ad the fomla fo calclatig the cost of a ode. Calclate the chage fo m The pogessio is,,. The cost of batches is : S { a ( ) d )} Fo m a d - S { a ( ) d )} ()() ( )( ) S { 9} 5 The chage is 5 { } SELF ASSESSMENT EXERCISE No.. Evalate the fist 5 tems of the pogessio 5 8. Evalate the fist tems of the pogessio - - 8 -.. Evalate the fist tems of the pogessio - 9-8... How may tems ae eqied i qestio () to podce a aswe of zeo? 5. Evalate the fist 5 tems of the pogessio give i ().. A mafacte of a mass podced podct chages by the followig method. Qatity Pice - - - ad so o with a maimm ode of How mch wold a ode of cost? (Aswe 9) (Aswe 57) (Aswe ) (Aswe ) (Aswe -5) (Aswe ) D.J.D www.feestdy.co.k

GEOMETRIC A geometic pogessio is a seies of mbes statig at a ad the et tem is the pevios tem mltiplied by a commo atio. It follows that the pogessio is a, a, a, a..a If we add tems togethe we have a GEOMETRIC SERIES. Fo eample let a. The two gaphs show the affect of havig slightly lage ad slightly smalle tha. Whe is smalle tha the gaph shows that S eaches a fial vale bt whe is lage tha, S keeps gowig. The sm of tems wold be give by: S a a a a. a S a(. ) mltiply both sides by ( - ) ( - )S a(. ) ( - ) ( - )S a{(. ) - (. )} ( - )S a{ - )} a{ } S Note this does ot wok if Sometimes we wat to kow the vale whe is ifiity. This is ot always possible bt ofte the tems will geeally get smalle ad smalle ad covege o a vale. This oly happes if the commo atio is less tha ad lage tha -. a Whe this happes the sm is S The poof is ot give hee. WORKED EXAMPLE No. Fid the sm of the fist tems of the geometic pogessio 5, 5(), 5(), 5()..5() a 5, ad a S a { } 5{ } 5 D.J.D www.feestdy.co.k

WORKED EXAMPLE No. 5 Fid the sm of the fist tems of the geometic pogessio, (), (), ()..() a, ad a S a { } { } 7 WORKED EXAMPLE No. Fid the sm to ifiity of the seies, ( ½), (½ ), (½)..(½) a, ½ a S a / / SELF ASSESSMENT EXERCISE No.. Evalate the fist tems of the seies whe (Aswe ). Evalate the maimm vale of the seies whe ½ (Aswe ). Evalate the maimm vale of the seies whe -½ (Aswe.) D.J.D www.feestdy.co.k 5

THE USE OF THE SUMMATION SYMBOL AND MORE ADVANCED SERIES May epessios ca be epeseted by a seies of mbes added togethe. This might be a seies of mbes like... 5 It might be a powe seies of like y... I geeal a seies may be witte as... whee is the tem i the seies. Coside the followig calclatio.... 5 The vale of is the sm of a seies of factios with the meatos fomig a descedig seies of iteges (whole mbes). We cold wite this moe simply as: o The symbol is a capital lette Sigma ad meas the sm of The limits of the vaiable betwee the smmatio takes place is show. WORKED EXAMPLE No. 7 Wite ot the seies epeseted by fo the fist tems. 8 8 9 Hece...... 9 WORKED EXAMPLE No. 8 Wite ot the seies epeseted by ( )!! ( ) ( ) ( ) ( )! ()() fo the fist tems.!!! ( ) ( ) ( ) ( )! Hece ( )...( )... ()()() D.J.D www.feestdy.co.k

LIMITING VALUES Coside the followig eqatio. Sppose we wish to kow if this has a vale 7 whe. A simple way to fid ot is to make se of the fact that / If we eaage the eqatio by dividig thogh by the highest ode of ( i this case) we get / / 7 / / Now pt We wite this as so thee is a limitig vale whe 7 7 5 L t 7 WORKED EXAMPLE No. 9 Fid the limitig vale of the followig epessios whe i) ii) ( )(5 7) 5 5 iii) i) / / ii) ( )(5 7) / ( /)( 5 7/) 5 iii) 5 5 5/ 5/ / SELF ASSESSMENT EXERCISE No.. Wite ot the seies epeseted by ( ) 5 - Aswe...( ) 5 7 fo the fist tems. ( -)! - ( -)!. Fid the limitig vale of the followig epessios whe 5 - ( )( ) i. 5 ii. iii. ( )( ) Aswes 5,.5 ad D.J.D www.feestdy.co.k 7

CONVERGENCE ad DIVERGENCE I the seies... we might thik that sice each tem is smalle tha the 5 oe befoe it, the the vale of wold ted to covege o some fige as we add moe ad moe tems. I fact the vale of will go o gettig bigge so i the limit as the vale of will also ted to ifiity. This seies has o limitig vale. We mst be vey caefl dealig with seies becase the vale of each tem may get bigge ad bigge (divegece) o it might get smalle ad smalle (covegece) ad if it coveges thee is a limitig vale. I ode to fid the vale of a seies whe we wite the seies i the fom:... o If the tem at is ot zeo the it seems likely that seies has o limitig vale ad is diveget. If L the seies might covege to a limitig vale bt this is ot cetai. t Fo eample coside agai the followig seies i. The seies is... L t It was show ealie that this seies i diveget so havig a zeo vale does ot pove the seies is coveget. WORKED EXAMPLE No. 9 Detemie if the followig seies is diveget. The seies is 5 8... / Lt ad sice this is ot zeo the seies is diveget. / D'ALEMBERT'S RATIO Fo ay seies L t < the seies coveges. t L > the seies diveges. This does ot tell s what happes if the eslt is ity. D.J.D www.feestdy.co.k 8

WORKED EXAMPLE No. Detemie if the followig seies is diveget. 5 7 The seies is... ( ) We ote that ad - ( )( ) ( ) Lt ( )( ) ( ) ( ) - ( ) ( ) ( /) ( /) This is less tha so the seies is coveget SELF ASSESSMENT EXERCISE No. Fid the limitig vale of the followig epessios whe. Detemie if the followig ae diveget o coveget. i. ii) 9 Aswes i. t L hece diveget. ii. t L hece idetemiate bt L ad sice this is ot zeo it mst be diveget. t D.J.D www.feestdy.co.k 9

ABSOLUTE CONVERGENCE Withot poof it ca be show that if we detemie the sm of the modlli of each tem i a seies sch that S... ad if this is coveget, the is also coveget ad has a defiite vale hece the se of the wods absolte covegece. It follows that if all the tems ae positive ayway, the if the seies is coveget it is absoltely coveget WORKED EXAMPLE No. 5 7 Test the followig seies to see if it has absolte covegece.... 5 7... This is the same as eample whece t L ( ) 5 7 This is less tha so the seies is coveget ad the seies... is absoltely coveget. WORKED EXAMPLE No. Detemie if the followig powe seies is coveget o diveget....!!!!! ( )! ( )! t L fo all vales of This is less tha so the seies is absoltely coveget fo all vales of. D.J.D www.feestdy.co.k

SELF ASSESSMENT EXERCISE No. 5 Test the followig seies fo covegece.!! 5 7...! 5! 7!....( ). ( ) Aswes! - ( -)!. L t so if < the seies is absoltely coveget bt if >it is diveget.. L t hece the seies is absoltely coveget fo all. FACTORIALS Yo shold aleady kow that a factoial mbe is deoted with! so! meas factoial.! ()()()()!()(-)(-)(-)... Withot poof,! Is always take as PASCAL'S TRIANGLE ( -)( - )( -)...( -{ }) A impotat factoial epessio is C! The top lie is the fist factos of ad the bottom lie is factoial If we evalated all the vales of C fom to we wold fid the vales ae symmetical. Fo eample take the case 5 5 5 5 5 (5)() 5 (5)()() 5 (5)()()() C C 5 C C C 5 ()() ()()() ()()()() 5 (5)()()()() C 5 (5)()()()() Pascal's Tiagle is made of ows as show. The th ow is made of all the mbes C fo to The zeo ow is C The st ow is C ad C The d ow is C C C The d ow is C C C C D.J.D www.feestdy.co.k

BINOMIAL THEOREM Coside a polyomial made p of epeated factos of (y ) If the (y) y y If the (y) y y y If the (y) y y y y If 5 the (y) y 5 5y y y 5 y 5 Let y. If the ( ) If the ( ) If the ( ) If 5 the ( ) 5 5 5 This is a seies with each tem cotaiig to a powe i ascedig ode fom to. Eamie this ad yo will see that the coefficiets match the ows of Pascal's Tiagle. It follows that the coefficiets ae C. Sice o ad we omally wite as simply the eqatio is moe commoly witte as: ( )... Notig that C C ( ) C C C C C... C ( ) C C C C... ( ) ( C ) Retig to ( y) we ca see that this is a seies sch that : ( y) C y C - y - C - y - C - y -... C ( y) y C - y - C - y - C - y -... This eslt ca be obtaied epadig ( ) as show i the et eample. WORKED EXAMPLE No. Usig the biomial theoem epad S (z a) Fist we mst eaage the epessio ito a fom that ca be epaded. a (z a) z let a/z ( ) (z a) z z Epadig we get S z ( ) z [ C C C C... ] S z ( ) z [ C (a/z) C (a/z) C (a/z) C (a/z)... (a/z) ] S z ( ) z [ C z - a C z - a C z - a C z - a... z - a ] S z ( ) z C z - a C z - a C z - a C z - a... a Note if we chage z to y ad a to we get the elt fo epadig ( y) y C y - C y - C y - C y -... D.J.D www.feestdy.co.k

WORKED EXAMPLE No. Epad ( ) with the biomial theoem. ( ) ( ) C we kow that ( ) C C C C C C C C... as solved ealie C 5 C C C C5 5 ( ) 5 5 Check this ot by pttig i ay vale of say () 79 5 ( ) ()() (5) () (5) () 79 9 79 79 WORKED EXAMPLE No. 5 Usig the biomial theoem epad ( ) aswe is - - - - ( ) C C C C... ( ) - - - ( )( ) ()() 5 5 C C y ad show that if is vey small the the - ( )( )( 5) ()()()... ( )... The eslt is a ifiite seies ad it is oly sefl fo evalatio whe is small sch that highe powes ae egligible. Check if. the y. -.9..9 WORKED EXAMPLE No. Usig the biomial theoem epad y ad evalate whe. y ( ) pttig - we ca epad. y ( ) C C C... ( )( ) ( )( )( ) y... ()() ()()() y... y...8....98. Note fo small vales of y is qite accately give by - D.J.D www.feestdy.co.k

SELF ASSESSMENT EXERCISE No.. Epad y ( ) sig the biomial theoem.. Epad y ( p) - to fo tems sig the biomial theoem.. Epad y ( q) - to fo tems sig the biomial theoem.. Epad y ad show that fo small mbes y 5. Epad ( ) y ad show that fo small mbes y 9 MACLAURIN'S SERIES This is a seies that demads special attetio ad the stdet might also look p the Taylo Seies o which it is based. The seies is based o sccessive diffeetials. f''() f'''() f f() f() f'()...!! WORKED EXAMPLE No. 7 ()!... Epad ito a seies sig Maclai's method. f() f() f () f () ( ) Hece...!!... f () f () ( )... bt clealy if the eslt is ifiity f () f () ( ) D.J.D www.feestdy.co.k

SELF ASSESSMENT EXERCISE No. 7 Epad the followig Maclai's method theoem. i. cos() ii. si() iii. e iv. sih() Check yo aswes i the table below. TABLE OF FUNCTIONS AND THEIR SERIES cos() si() cosh() sih() e l( ) ( )! ( ) ( ) ( )! ( )!! ( )! ( ) D.J.D www.feestdy.co.k 5

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