1. 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.

2. Find lower and upper bounds for the sum of i.e

3. 1234 n y = 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+1 1 2

4. n–1 n 1234 n y = 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

5. n–1 n 1234 n y = n+1 Represent the series by a set of rectangles, and approximate the area of these rectangles by the area under a curve. The area under y = from x = 1 to x = n+1 is too large 1 2

6. n–1 n 1234 n y = 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 So from eqn (1)

7. n–1 n 1234 n y = 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 So the sum of the areas of the rectangles still represents the sum of the series.

8. n–1 n 1234 n y = n+1 To obtain a lower bound, use the same rectangles translated 1 unit to the left. 1 2 The area under y = from x = 0 to x = n is too small

9. n–1 n 1234 n y = n+1 To obtain a lower bound, use the same rectangles translated 1 unit to the left. 12 So

10. Use the graph of y = To find upper and lower bounds for the sum of the series  +  +  +  … You do this one! n–1 nn+1 1 2 Lower Bound

11. Use the graph of y = To find upper and lower bounds for the sum of the series  +  +  +  … You do this one! n–1n 1 2 Upper Bound

12. Use the graph of y = x 2 To find upper and lower bounds for the sum of the series 1 2 +2 2 +3 2 +4 2 …10 2 You do this one!

13. Use the graph of y = x 2 To find upper and lower bounds for the sum of the series 1 2 +2 2 +3 2 +4 2 …10 2 You do this one! Upper Bound

14. Use the graph of y = x 2 To find upper and lower bounds for the sum of the series 1 2 +2 2 +3 2 +4 2 …10 2 You do this one! Lower Bound