Calculus Summation Notation
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S
Summation Notation

 

 Math is like a foreign language, the symbols all stand for words and ideas.

 The S symbol, Greek letter capital Sigma, has a fairly involved meaning.
 It tells us to sum up a bunch of terms.
 It is kind of like a little computer program.
 
 How many terms are we talkin' about?
 
 That bit of info comes from other symbols that are above and below the S.
 You might see something like:

 

 

 Maybe instead of the X you will see a Y or a k or any old letter below the S.
 The important thing is that if you count from the bottom number (here a 1), 
 to the top number (here a 5) you get the number of terms we add together.
 

 1, 2, 3, 4, 5 (5 terms)

 
 If we had:

 

 We would also have 5 terms.

  103, 104, 105, 106, 107 (5 terms)

 
 Or:

 

 Which would be:

 -4, -3, -2, -1, 0 (5 terms)

 
 WHY THIS SILLY NOTATION?
 If you wanted 5 terms, why not just write "5 terms" under the S or something?
 
 The reason is that those particular numbers:
 1, 2, 3, 4, 5 or 103, 104, 105, 106, 107
 might be needed somewhere in the problem.
 Where we wrote (stuff), 
 we might have something like:

 3X

 This makes the whole expression:

 

 Did you see that the letter under the S  
 matches the variable in the expression to the right?
 This little group of symbols means that we have 5 terms.
 Each of the terms is the number 3, raised to a power.
 The power is the sequence of numbers.
 
 This particular whole sum is:

3+ 32 + 33 + 34 + 35 = 363

 
 Which is a whole lot different from:

 

 Which is:

 3103  + 3104 + 3105 + 3106 + 3107 1.684 x 1051

 
 Sometimes there is no variable in the statement.
 We might have:

 

 Which is just:

4 + 4 + 4 + 4 + 4 + 4 = 24

 We have a total of 6 fours here.
 That's the same as 6 x 4.
 so we can say that if we're "summing" a term with no variable 
 and the number under the S is a 1
 the answer is the number on top times the constant.
 In math talk, that is:

 

 So:

 

 
 If we have something like:

 

 
 We get:

3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 45

 
 Remember when we did limits, and derivatives, and integrals?
 The rules let us move constants to the left and deal with them later.
 We can do that here too.
 So:

 

 
 Also, when we worked with limits, and derivatives, and integrals
 and had a bunch of terms, like:

 3X2 + 4X - 5

 
 We could work with each term by itself.
 We can do that here too.
 So:

 

 
 Even so, calculating this out would take a long time.
 Just the first part would be:

 3(1+ 22 + 32 + 42 + 52)

 
 And these things can get even more complex.
 How about some shortcuts?
 SURE!
 
 If we have X to the first power:

 

 
 If we have X to the second power (X 2):

 

 
 If we have X to the third power (X 3):

 

 
 If we have some number to the X power (C X):

 

 

   copyright 2005 Bruce Kirkpatrick

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