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Sigma notation
mc-TY-sigma-2009-1 Sigma notation is a method used to write out a long sum in a concise way. In this unit we look at ways of using sigma notation, and establish some useful rules. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: expand a sum given in sigma notation into an explicit sum; write an explicit sum in sigma notation where there is an obvious pattern to the individual terms; use rules to manipulate sums expressed in sigma notation.
Contents
1.Introduction2
2.Some examples3
3.Writing a long sum in sigma notation 5
4.Rules for use with sigma notation6
www.mathcentre.ac.uk 1c?mathcentre 2009
1. IntroductionSigma notation is a concise and convenient way to represent long sums. For example, we often
wish to sum a number of terms such as
1 + 2 + 3 + 4 + 5
or
1 + 4 + 9 + 16 + 25 + 36
where there is an obvious pattern to the numbers involved. The first of these is the sum of the first five whole numbers, and the second is the sum of the first six square numbers. More generally, if we take a sequence of numbersu1,u2,u3,...,unthen we can write the sum of these numbers as u
1+u2+u3+...+un.
A shorter way of writing this is to leturrepresent the general term of the sequence and put n r=1u r. Here, the symbolΣis the Greek capital letterSigmacorresponding to our letter 'S", and refers to the initial letter of the word 'Sum". So this expression means the sum of all the termsur wherertakes the values from 1 ton. We can also write b r=au r to mean the sum of all the termsurwherertakes the values fromatob. In such a sum,ais called the lower limit andbthe upper limit.
Key Point
The sumu1+u2+u3+...+unis written in sigma notation as n r=1u r. www.mathcentre.ac.uk 2c?mathcentre 2009
Exercises1. Write out what is meant by
(a) 5? n=1n
3(b)5?
n=13 n(c)4? r=1(-1)rr2(d)4? k=1(-1)k+1 2k+ 1 (e) N? i=1x 2 i(f)N? i=1f ixi
2. Evaluate
4? k=1k 2.
2. Some examples
Example
Evaluate
4? r=1r 3.
Solution
This is the sum of all ther3terms fromr= 1tor= 4. So we take each value ofr, work out r
3in each case, and add the results. Therefore
4 r=1r
3= 13+ 23+ 33+ 43
= 1 + 8 + 27 + 64 = 100.
Example
Evaluate
5? n=2n 2.
Solution
In this example we have used the letternto represent the variable in the sum, rather thanr. Any letter can be used, and we find the answer in the same way as before: 5 n=2n
2= 22+ 32+ 42+ 52
= 4 + 9 + 16 + 25 = 54.
Example
Evaluate
5? k=02 k. www.mathcentre.ac.uk 3c?mathcentre 2009 SolutionNotice that, in this example, there are 6 terms in the sum, because we havek= 0for the first term: 5 k=02 k= 20+ 21+ 22+ 23+ 24+ 25 = 1 + 2 + 4 + 8 + 16 + 32 = 63.
Example
Evaluate
6? r=11
2r(r+ 1).
Solution
You might recognise that each number
1
2r(r+ 1)is atriangular number, and so this example
asks for the sum of the first six triangular numbers. We get 6 r=11
2r(r+ 1) =?12×1×2?+?12×2×3?+?12×3×4?+?12×4×5?
?1
2×5×6?+?12×6×7?
= 1 + 3 + 6 + 10 + 15 + 21 = 56.
What would we do if we were asked to evaluate
n? k=12 k? Now we know what this expression means, because it is the sum of all the terms2kwherek takes the values from 1 ton, and so it is n k=12 k= 21+ 22+ 23+ 24+...+ 2n. But we cannot give a numerical answer, as we do not know the value of the upper limitn.
Example
Evaluate4?
r=1(-1)r.
Solution
Here, we need to remember that(-1)2= +1,(-1)3=-1, and so on. So 4 r=1(-1)r= (-1)1+ (-1)2+ (-1)3+ (-1)4 = (-1) + 1 + (-1) + 1 = 0. www.mathcentre.ac.uk 4c?mathcentre 2009
ExampleEvaluate3?
k=1? -1 k? 2quotesdbs_dbs2.pdfusesText_2
Sigma notation - mathcentreacuk