Calculus Summation Notation
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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)



 Which would be:

 -4, -3, -2, -1, 0 (5 terms)

 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:


 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:




 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.


 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.


 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?
 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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