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

6.3 Series 213
whereS ∞
is known as the ‘sum to infinity’. Ifr > 1 orr < −1,r k fails to approach a
finitelimitask → ∞and the geometric series diverges.
Sum of aninfinite geometric series:S ∞
=
a
1 −r
−1<r<1
Example6.14 Findthe sum toktermsofthe following series and deducetheirsumstoinfinity:
(a) 1+ 1 3 + 1 9 + 1
27 +··· (b) 12+6+3+11 2 +···
Solution (a) This isageometric series with first term1and common ratio1/3.Therefore,
S k
= a(1−rk )
= 1(1 − (1/3)k )
= 3 ( ( 1 k )
1 −
1 −r 2/3 2 3)
Ask → ∞, (1/3) k → 0sothatS ∞
= 3/2.
(b) This isageometric series with first term12 and common ratio 1 2 . Therefore,
S k
= 24(1 − (1/2) k )
Ask → ∞, (1/2) k → 0sothatS ∞
= 24.Thiscould,ofcourse,havebeenobtained
directly from the formulafor the sum toinfinity.
EXERCISES6.3
1 An arithmeticseries has afirstterm of4andits 30th
term is1000.Find the sum to 30terms.
2 Find the sum to 20 termsofthe arithmeticserieswith
firstterma,andcommon differenced,given by
(a)a=4,d=3 (b)a=4,d=−3
3 Ifthe sumto 10terms ofan arithmeticseries is100
anditscommondifference,d,is −3,finditsfirstterm.
4 Thesum to 20terms ofan arithmeticseries is
identicalto the sum to 22 terms. Ifthe common
difference is −2, find the firstterm.
5 Find the sum to five terms ofthe geometricseries
with firstterm 1 andcommon ratio 1/3. Find the sum
to infinity.
6 Find the sum ofthe first20terms ofthe geometric
serieswith firstterm 3 andcommon ratio 1.5.
7 Find the sum ofthe arithmeticseries with firstterm 2,
common difference 2, andlast term50.
8 The sumto infinityofageometricseries isfour times
the firstterm.Find the common ratio.
9 The sumto infinityofageometricseries istwicethe
sum ofthe firsttwo terms. Findpossiblevalues ofthe
common ratio.
10 Express the alternatingharmonic series
1 − 1 2 + 1 3 − 1 + · · ·in sigma notation.
4
11 Writedown the firstsixterms ofthe series ∑ ∞
k=0 z −k .
12 Explain why ∑ ∞
k=1 x[k] is the same as ∑ ∞
n=1 x[n].
Further, explain why both canbe written as
∑ ∞k=0
x[k +1].
Solutions
1 15060
2 (a) 650 (b) −490
3 23.5
4 41