Page 178 - Calculus Workbook For Dummies

162 Part IV: Integration and Infinite Series

Sigma Notation and Reimann

Sums: Geek Stuff

Now that you’re warmed up, let’s segue into summing some sophisticated sigma sums.
Sigma notation may look difficult, but it’s really just a shorthand way of writing a
long sum.

In a sigma sum problem, you can pull anything through the sigma symbol to the out-
side except for a function of the index of summation (the i in the following example).
Note that you can use any letter you like for the index of summation, though i and k
are customary.

12
Q. Evaluate ! i 5 . 2. Set the range of the sum.
2
i = 4 Ask yourself what i must be to make the
A. The sum is 3180. first term equal 50 — that’s 5, of course.
3
And ask the same question about the last
1. Pull the 5 through the sigma symbol. 3
term of 150 — that gives you i = 15:
12
15
5 i ! 2 !^ 10 ih 3
i = 4
i = 5
2. Plug 4 into i, then 5, then 6, and so on 3. Simplify.
up to 12, adding up all the terms.
15
=! 10 3 i 3
2
2
2
2
2
2
2
2
5 4 +
= _ 2 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 i
i = 5
15
3. Finish on your calculator. = 1000 i ! 3
i = 5
5 636 =
= ^ h 3180
4. (Optional) Set the i to begin at zero or
Q. Express 50 + 60 + 70 + 80 + ... 150 with one.
3
3
3
3
3
+
sigma notation. It’s often desirable to have i begin at 0
11
A. 1000!^ i + 4h 3 or 1. To turn the 5 into a 1, you subtract
4. Then subtract 4 from the 15 as well.
=
i 1
To compensate for this subtraction, you
1. Create the argument of the sigma add 4 to the i in the argument:
function.
11
3
The jump amount between terms in a long = 1000!^ i + 4h
i 1
=
sum will become the coefficient of the
index of summation in a sigma sum, so
you know that 10i is the basic term of your
argument. You want to cube each term, so
that gives you the following argument.
3
!^ 10 ih