Sigma notation
(?1)k 1 k . Key Point. To write a sum in sigma notation try to find a formula involving a variable k where the first.
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INTRODUCTION TO SIGMA NOTATION 1. The notation itself Sigma
The variable k is called the index of the sum. The numbers at the top and bottom of the ? are called the upper and lower limits of the summation. In this case
1 Convergence Tests
Root Test and Ratio Test. The root test is used only if powers are involved. Root Test. ? k2. 2k converges: (ak). 1/k. =
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Feb 12 2006 b) P(k) ? P(k + 1) for all natural numbers k . The standard analogy to this involves a row of dominoes: if it is shown.
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Remarque : • La variable k est une variable muette c'est à dire qu'une fois la somme calculée le résultat ne dépend plus de k
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uk la somme de la série ?k?0 uk et on dit que la série est convergente Sinon on dit qu'elle est divergente Notations On peut noter une série de
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Le symbole ? (sigma) s'utilise pour désigner de manière générale la somme de plusieurs termes Ce symbole est généralement accompagné d'un indice que l'on
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1 Exercice 5 : Somme de termes en progression arithmétique — Soit (uk) une suite de nombres réels en progression arithmétique Soit(
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un changement où on inverse l'ordre d'énumération : on pose l = n ? k ?? k = n ? l Après un changement d'indice le nombre de termes dans la somme doit
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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
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18 sept 2010 · Exemple 1 : Calcul de la somme des entiers • Nous allons démontrer par récurrence que la propriété Pn : i=n ?
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Déployer une somme Quand je parlerai de déployer une somme cela signi era qu'on récrit une somme initialement présentée avec le symbole sigma n ? k=1
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(Voir la correction ici) Calculer les sommes suivantes : 1 S = 10 ? k=1 k k=1 k2 9 D = n+2 ? k=5 k 10 E = 2n ? k=n k Exercice 2
![[PDF] Sigma notation - Mathcentre](https://pdfprof.com/Listes/18/18022-18mc-ty-sigma-2009-1.pdf.pdf.jpg "[PDF] Sigma notation - Mathcentre")
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 20091. IntroductionSigma notation is a concise and convenient way to represent long sums. For example, we often
wish to sum a number of terms such as1 + 2 + 3 + 4 + 5
or1 + 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 u1+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 2009Exercises1. Write out what is meant by
(a) 5? n=1n3(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 ixi2. 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 r3in each case, and add the results. Therefore
4 r=1r3= 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=2n2= 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=112r(r+ 1).
Solution
You might recognise that each number
12r(r+ 1)is atriangular number, and so this example
asks for the sum of the first six triangular numbers. We get 6 r=112r(r+ 1) =?12×1×2?+?12×2×3?+?12×3×4?+?12×4×5?
?12×5×6?+?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 2009ExampleEvaluate3?
k=1? -1 k? 2Solution
Once again, we must remember how to deal with powers of-1: 3 k=1? -1 k? 2 =?-11?2+?-12?
2+?-13?
2 = 1 + 1 4+19 = 1 13 36.3. Writing a long sum in sigma notation
Suppose that we are given a long sum and we want to express it insigma notation. How should we do this? Let us take the two sums we started with. If we want to write thesum1 + 2 + 3 + 4 + 5
in sigma notation, we notice that the general term is justkand that there are 5 terms, so we would write1 + 2 + 3 + 4 + 5 =5?
k=1k .To write the second sum
1 + 4 + 9 + 16 + 25 + 36
in sigma notation, we notice that the general term isk2and that there are 6 terms, so we would write1 + 4 + 9 + 16 + 25 + 36 =6?
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