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

6.3.2 Sumofafinitegeometricseries
6.3 Series 211
A geometric series is the sum of the terms of a geometric progression. If we sum the
geometric progression 1, 1 2 , 1 4 , 1 8 , 1
16 wefind
S=1+ 1 2 + 1 4 + 1 8 + 1
16
(6.1)
Iftherehadbeenalargenumberoftermsitwouldhavebeenimpracticaltoaddthemall
directly. However, let us multiplyEquation (6.1) by the common ratio, 1 2 :
1
2 S = 1 2 + 1 4 + 1 8 + 1
16 + 1
32
so that, subtracting Equation (6.2)from Equation (6.1), we find
(6.2)
S − 1 2 S=1− 1
32
since mostterms cancel out. Therefore 1 2 S = 31
32 andsoS=31 16 = 115 16 .
We can apply this approach more generally: when we have a geometric progression
with first termaand common ratior, the sum toktermsis
S k
=a+ar+ar 2 +ar 3 +···+ar k−1
Multiplying byr gives
rS k
=ar+ar 2 +ar 3 +···+ar k−1 +ar k
Subtraction givesS k
−rS k
=a −ar k , so that
S k
= a(1−rk )
1 −r
providedr ≠ 1
This formula gives the sum to k terms of the geometric series with first term a and
common ratior.
Sumofageometricseries:S k
= a(1−rk )
1 −r
r≠1
6.3.3 Sumofaninfiniteseries
When dealing with infinite series the situation is more complicated. Nevertheless, it is
frequently the case that the answer to many problems can be expressed as an infinite
series.Incertaincircumstances,thesumofaseriestendstoafiniteanswerasmoreand
more terms are included and we say the series has converged. To illustrate this idea,
consider the graphical interpretation of the series 1 + 1 2 + 1 4 + 1 8 +···,asgivenin
Figure 6.6.
Start at0and move a length 1:total distance moved = 1
Move further, a length 1 2 : total distance moved = 11 2
Move further, a length 1 4 : total distance moved = 13 4