BC 3
Taylor Series
Name:
Estimating Error, Part 1


Recall that if the alternating series
 1  ak  or
k 1

must be the case that S  Sn  an1 for all n  1.

k 1


 1k  ak 
k 1
converges to a value S, then it

1.
Find an upper bound for the error made in approximating sin  3.5 using the first six nonzero
terms of its Maclaurin series.
2.
What are the possible values for x if we want the error in approximating sin  x  using the first six
nonzero terms of its Maclaurin series to be less than 0.0005?
3.
Find an upper bound for the error made in approximating tan 1  0.8  using the first seven
nonzero terms of its Maclaurin series.
Taylor Series 1.1
4.
When approximating tan 1  0.8  , how many nonzero terms of its Maclaurin series do you have to
use in order to have an error that is less than 0.00001?
5.
What are the possible values for x if we want the error in approximating cos  x  using the first six
nonzero terms of its Maclaurin series to be less than 0.0005?
6.
When approximating cos  2.75 , how many nonzero terms of its Maclaurin series do you have to
use in order to have an error that is less than 0.00001?
7.
If x > 0, what are the possible values for x if we want the error in approximating ln 1  x  using
the first seven nonzero terms of its Maclaurin series to be less than 0.0005?
Taylor Series 1.2
8.
When approximating ln 1.75 , how many nonzero terms of its Maclaurin series do you have to
use in order to have an error that is less than 0.00001?
1
2
9.
Find an upper bound for the error made in approximating e
its Maclaurin series?
10.
If x < 0, what are the possible values for x if we want the error in approximating e x using the first
seven nonzero terms of its Maclaurin series to be less than 0.0005?
11.
When approximating e4.1 , how many nonzero terms of its Maclaurin series do you have to use in
order to have an error that is less than 0.00001?
Taylor Series 1.3
using the first six nonzero terms of

12.
13.

xn
Suppose you are given the power series
.
1 n
n 0
a.
If x < 0, what are the possible values for x if we want the error in approximating the value of
the series using the first 15 nonzero terms of the series to be less than 0.0005?
b.
When approximating the value of the series for x = –0.9, how many nonzero terms of the
series do you have to use in order to have an error that is less than 0.0001?
In problems 2, 5, 7, 10, and 12a, you had to determine the possible values for x that could be used
in a specified approximation. In problems 2 and 5, no stipulation about the value of x was
included. But in problems 7, 10, and 12a, stipulations were placed on the value of x (either x > 0 or
x < 0). Explain why these stipulations were necessary?
Taylor Series 1.4