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Using Areas to Approximate to Sums of Series
In the previous examples it was shown that an area can be represented by the sum of a series.
Conversely the sum of a series can be represented by an area.
1 2 3 ....... n n
1
r
Find lower and upper bounds for the sum of
i.e
1 2 3 4 n
y = x
Represent the series by a set of rectangles, and approximate the area of these rectangles by the area under a curve.
Rectangle 1 has an area of 1 x 1 = 1
Rectangle 2 has an area of 1 x 2 = 2
Rectangle n has an area of 1 x n = n
n–1 n n+11 2
n–1 n
1 2 3 4 n
y = x
n+1
Represent the series by a set of rectangles, and approximate the area of these rectangles by the area under a curve.
Hence the sum of the rectangle areas =
1 2 3 ....... n
1 2
n–1 n
1 2 3 4 n
y = x
n+1
Represent the series by a set of rectangles, and approximate the area of these rectangles by the area under a curve.
n 1n
1 1
So r x dx
The area under y = x from x = 1 to x = n+1 is too large
1 2
n–1 n
1 2 3 4 n
y = x
n+1
Represent the series by a set of rectangles, and approximate the area of these rectangles by the area under a curve.
1 2
n 1n 1 3 3
2 2
1 1
2 2 2xdx x n 1
3 3 3
n 3
2
1
2 2r n 1
3 3 So from eqn (1)
n–1 n
1 2 3 4 n
y = x
n+1
To obtain a lower bound, use the same rectangles translated 1 unit to the left.
1 2
but the x limits will be from 0 to n.
Each rectangle still has a width of 1 and a height of x
So the sum of the areas of the rectangles still represents the sum of the series.
n–1 n
1 2 3 4 n
y = x
n+1
To obtain a lower bound, use the same rectangles translated 1 unit to the left.
1 2
nn
1 0
So r x dx
The area under y = x from x = 0 to x = n is too small
n–1 n
1 2 3 4 n
y = x
n+1
To obtain a lower bound, use the same rectangles translated 1 unit to the left.
1 2
nn 3 3
2 2
0 0
2 2xdx x n
3 3
n3 3
2 2
1
2 2 2n r n 1
3 3 3 So
Use the graph of y = 1x
To find upper and lower bounds for the sum of the series +++…
n1
-1
1
0X->
|̂Y
You do this one!
n–1 n n+11 2
Lower Bound
Use the graph of y = 1x
To find upper and lower bounds for the sum of the series +++…
n1
-1
1
0X->
|̂Y
You do this one!
n–1 n1 2
Upper Bound
Use the graph of y = x2
To find upper and lower bounds for the sum of the series 12+22+32+42…102
You do this one!
-10
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50
60
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80
90
100
1 2 3 4 5 6 7 8 9 100X->
|̂Y
Use the graph of y = x2
To find upper and lower bounds for the sum of the series 12+22+32+42…102
You do this one!
-10
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 100X->
|̂Y
Upper Bound
Use the graph of y = x2
To find upper and lower bounds for the sum of the series 12+22+32+42…102
You do this one!
-10
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 100X->
|̂Y
Lower Bound
