The properties associated with the summation process are given in the following rule.
Rule: properties of sigma notation
Let
and
represent two sequences of terms and let
c be a constant. The following properties hold for all positive integers
n and for integers
m , with
Proof
We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises.
2. We have
3. We have
□
A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for
sums and powers of integers , and we use them in the next set of examples.
Rule: sums and powers of integers
The sum of
n integers is given by
The sum of consecutive integers squared is given by
The sum of consecutive integers cubed is given by
Evaluation using sigma notation
Write using sigma notation and evaluate:
The sum of the terms
for
The sum of the terms
for
Multiplying out
we can break the expression into three terms.
Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let
be a continuous, nonnegative function defined on the closed interval
We want to approximate the area
A bounded by
above, the
x -axis below, the line
on the left, and the line
on the right (
[link] ).
How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval
into
n subintervals of equal width,
We do this by selecting equally spaced points
with
and
for
We denote the width of each subinterval with the notation Δ
x , so
and
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