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INTRODUCTION TO SIGMA NOTATION - Ohio State University

INTRODUCTION TO SIGMA NOTATION

1.The notation itselfSigmanotation is a way of writing a sum of many terms, in a concise

form. A sum in sigma notation looks something like this: 5 X k=13k The (sigma) indicates that a sum is being taken. The variablek is called theindex of the sum. The numbers at the top and bottom of the are called theupperandlowerlimits of the summation. In this case, the upper limit is 5, and the lower limit is 1. The notation means that we will take every integer value ofkbetween 1 and 5 (so

1, 2, 3, 4, and 5) and plug them each into the summand formula (here

that formula is 3k). Then those are all added together. 5 X k=13k= 31 + 32 + 33 + 34 + 35 = 45 Example 1.Write out what is meant by the following: 3 X k=01k+ 1 Here, the indexktakes the values 0, 1, 2, and 3. We'll plug those each into

1k+1and add them together.

3 X k=01k+ 1=10 + 1 +11 + 1 +12 + 1 +13 + 1 Example 2.Write out what is meant by the following: 8 X i=1(1)iDate: Friday March 31, 2016. 1

2 INTRODUCTION TO SIGMA NOTATION

The index variable here is written asiinstead ofk. That's ok. The most common variables to use for indexes includei,j,k,m, andn. 8X i=1(1)i= (1)1+ (1)2+ (1)3+ (1)4+ (1)5+ (1)6+ (1)7+ (1)8 =1 + 1 +1 + 1 +1 + 1 +1 + 1 = 0

Try one on your own.

Example 3.Write out what is meant by the following (no need to simplify): 4X n=1pn+ 1

Let's try going the other way around.

Example 4.Write the following sum in sigma notation.

2 + 4 + 6 + 8 +:::+ 22 + 24

Notice that we can factor a 2 out of each term to rewrite this sum as

21 + 22 + 23 + 24 +:::+ 211 + 212

That means that we are adding together 2 times every number between

1 and 12. The sigma notation could be

12X k=12k There is no need to usekas our index variable. We could have just as easily usedmorjinstead. 12X k=12k=12X m=12m=12X j=12j

Notice, that these are NOT the same as

12X k=12m Example 5.Write the following sum in sigma notation. 112
+14 18 +:::164 +1128

INTRODUCTION TO SIGMA NOTATION 3

This one is a little more complicated. We'll worry about the signs later, rst we'll deal with the numbers themselves. Do you notice a pattern in the terms? Sure, we get to the next term by dividing by 2.

That is:

1 = 12 0=12 0 12 =12 1=12 1 14 =12 2=12 2 1128
=12 7=12 7 If we call our index variablek, thenkshould go from 0 to 7, and the numbers themselves are just12 k. Now we need to deal with the signs. We say above that (1)kwill alternate between +1 and1. That is, if we multiply our terms from above by (1)k, they will alternate between + and. We are starting withk= 0, so (1)0= +1 will give us the alternation starting at the sign we want. 7X k=0(1)k12 k =7X k=0 12 k

Try one on your own.

Example 6.Write the following sum in sigma notation. 1p2 2p3 +3p4 4p5 +:::+51p52 52p53

2.Calculating with sigma notation

We want to use sigma notation to simplify our calculations. To do that, we will need to know some basic sums. First, let's talk about the sum of a constant. (Notice here, that our upper limit of summation isn.nis not the index variable, here, but the highest value that the index variable will take.) nX k=1C

4 INTRODUCTION TO SIGMA NOTATION

This is a sum ofnterms, each of them having a valueC. That is, we are addingncopies ofC. This sum is justnC. The other basic sums that we need are much more complicated to derive. Rather than explaining where they come from, we'll just give you a list of the nal formulas, that you should remember.

Formula 1.

nX k=1C=nC.

Formula 2.

nX k=1k=n(n+ 1)2

Formula 3.

nX k=1k

2=n(n+ 1)(2n+ 1)6

Formula 4.

nX k=1k

3=n(n+ 1)2

2 Now that we have this list, let's use them to compute.

Example 7.Find the value of the sum5X

k=19. This is just the sum of a constant, withC= 9 andn= 5. The value isnC= 59 = 45.

Example 8.Find the value of the sum100X

k=1k. This is the sum 1+2+3+:::+100. According to Formula 2 above (withn= 100), this is100(101)2 = 5050. Because sigma notation is just a new way of writing addition, the usual properties of addition still apply, but a couple of the important ones look a little dierent.

Property 1(Commutativity).nX

k=1(ak+bk) =nX k=1a k+nX k=1b k.

Property 2(Distribution).nX

k=1cak=cnX k=1a k.

Example 9.Find the value of the sum10X

k=1quotesdbs_dbs2.pdfusesText_2