Work Sheet 31—Estimating the Sum of a Series
NAME__________________________________________________
INSTRUCTIONS: You may refer to Section 11.3 Integral Test and Estimating
Sums to answer the following questions. You may discuss it with a classmate..
BASIC IDEA –To show how a partial sum can be used to obtain upper and lower bounds
on the sum of the series when the hypotheses of the Integral Test are satisfied.
By using an area argument and the following figure it is possible to show that
∞
+∞
∫n +1f ( x) dx <
∑
+∞
a k < ∫n f (x) dx
(1)
k =n +1
∞
Let S be the sum of the series
∑ ak
k =1
and snis the nth partial sum. Add sn to every part of
the above inequality
+∞
sn + ∫n+1 f (x) dx < sn +
∞
∑
+∞
a k < sn + ∫n f ( x) dx
(2)
k =n +1
The middle portion of the inequality is the sum of the series S.
+∞
+∞
sn + ∫n+1 f (x) dx < S < sn + ∫n f ( x) dx .
(3)
Let the error in approximating the exact S by the nth partial sum be Rn= S – sn. Subtract
sn from each portion of the inequality
+∞
∫n +1f ( x) dx
Question 1
+∞
< R n < ∫n f ( x) dx
∞
1
π2
It is known that ∑ 2 =
.
6
k =1 k
(a)
Show that if sn is the nth partial sum of this series, then
1
(4)
1
π2
1
sn +
<
< sn +
n +1 6
n
(b)
Calculate s3 exactly, and then use the result in part (a) to show that
29 π 2 61
<
<
18
6
36
(c)
Find upper and lower bounds on the error that results if the sum of the series is
approximated by the 10th partial sum.
Question 2
Find upper and lower bounds on the error that results if the sum of the series is
approximated by the 19th partial sum
∞
1
∑ (2k + 1)2
k =1
2
Question 3
∞
Our objective in this problem is to approximate the sum of the series
1
∑k
3
to two
k=1
decimal-place accuracy.
(a)
Show that if S is the sum of the series and sn is the nth partial sum then
1
1
sn +
2 < S < sn +
2n2
2(n + 1)
(b)
For two decimal-place accuracy, the error must be less than 0.005. Find the
smallest value of n that will be necessary to achieve this accuracy.
(c)
Calculate sn for the n obtained in part (b). Keep at least 5 decimal places.
(d)
Use the result of part (c) and part (a). Approximate the sum, S, by the midpoint of
the interval. Round to two decimal places.
3