082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

214 Chapter 6 Sequences and series
5 1.494,S ∞ = 3 2
6 19946
7 650
8
3
4
9 ± √ 1 2
∑
10 ∞ (−1) n+1
1 n
11 1+ 1 z + 1 z 2 + 1 z 3 + 1 z 4 + 1 z 5
TechnicalComputingExercises6.3
Most technicalcomputing languageshave built-in
functionsforgenerating geometricseries.MATLAB ®
has the functioncumprod whichcalculates the
cumulative productofthe numberspassed to its
input. Ittakes the first term andthensuccessively
multiplies the succeedingarguments in turnbythe
result.
Forexample, in MATLAB ® wecould type:
S=cumprod([1 0.5 0.5 0.5 0.5])
to produce the geometricseriesandstore itin a row
vectorS.
1.000000 0.500000 0.250000
0.125000 0.062500
1 Calculate the sum ofallofthe elementsin the finite
seriesgiven above.
2 Increase the number ofelementsin the series to 10
andnotethe difference in youranswer.
3 Compare the answers to the previous two exercises to
the exact equationgiven atthe endofSection6.3.2
witha = 1 andr = 0.5.Whatwould you expect to
happeniftherewere100 elementsin the series?
6.4 THEBINOMIALTHEOREM
It is straightforward to show that the expression (a + b) 2 can be written as a 2 +
2ab + b 2 . It is slightly more complicated to expand the expression (a + b) 3 to a 3 +
3a 2 b +3ab 2 +b 3 . However, it is often necessary to expand quantities such as (a +b) 6
or (a +b) 10 , say, and then the algebra becomes extremely lengthy. A simple technique
forexpandingexpressionsoftheform (a+b) n ,wherenisapositiveinteger,isgivenby
Pascal’s triangle.
Pascal’striangleisthetriangleofnumbersshowninFigure6.7,whereitisobserved
that every entry is obtained by adding the two entries on either side in the preceding
row, always starting and finishing a row with a ‘1’. You will note that the third row
down,1 2 1,givesthecoefficientsintheexpansionof (a +b) 2 = 1a 2 +2ab +1b 2 ,
whilethefourthrow,1 3 3 1,givesthecoefficientsintheexpansionof (a +b) 3 =
1a 3 +3a 2 b +3ab 2 +1b 3 .Furthermore,thetermsintheseexpansionsarecomposedof
decreasingpowersofaandincreasingpowersofb.Whenwecometoexpandthequantity
(a +b) 4 therowbeginning‘1 4’inthetrianglewillprovideuswiththenecessary
coefficientsintheexpansionandwemustsimplytakecaretoputinplacetheappropriate
powers ofaandb. Thus (a +b) 4 = 1a 4 +4a 3 b +6a 2 b 2 +4ab 3 +1b 4 .
1
1 1
1 2 1
1 3 3 1
1 4 + 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Figure6.7
Pascal’striangle.