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

6.3 Series 209
13 ±128
14 3
15 0
16 (a) Limitdoes not exist (b) 1 2
(c) 1
17 16
18 (a) Limit doesnot exist
(b) Limit doesnotexist
(c) 0 (d) 3 5
(e) 0
6.3 SERIES
Whenever the terms of a sequence are added together we obtain what is known as a
series.Forexample,ifweaddthetermsofthesequence1, 1 2 , 1 4 , 1 8 ,weobtaintheseries
S, where
S=1+ 1 2 + 1 4 + 1 8
This series ends after the fourth term and is said to be a finite series. Other series we
shall meet continue indefinitely and aresaidtobeinfinite series.
Given an arbitrarysequencex[k], we use thesigmanotation
n∑
S n
= x[k]
k=1
to mean the sum x[1] +x[2] + ··· +x[n], the first and last values of k being shown
below and above the Greek letter , which is pronounced ‘sigma’. If the first term of
the sequence isx[0] rather thanx[1] wewould write ∑ n
k=0 x[k].
6.3.1 Sumofafinitearithmeticseries
An arithmetic series isthe sum ofan arithmetic progression. Consider the sum
S=1+2+3+4+5
Clearly this sums to 15. When there are many more terms it is necessary to find a more
efficient way of adding them up. The equation forScan be writtenintwo ways:
and
S=1+2+3+4+5
S=5+4+3+2+1
If weadd these two equations together we get
2S=6+6+6+6+6
There arefive termsso that
thatis
2S=5×6=30
S=15
Now a general arithmeticseries withkterms can be writtenas
S k
=a+(a+d)+(a+2d)+···+(a+(k−1)d)
butrewritingthis back tofront, wehave
S k
=(a+(k−1)d)+(a+(k−2)d)+···+(a+d)+a