Question 05 (a)
Write down the first three nonzero terms obtained by using Maclaurin series to estimate , and explain why the error in using this estimate is less than 0.001.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
The Maclaurin series for is
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
As suggested in the hint, we know that
Plugging in gives
Now, integrating gives
Now, using just the first three terms and integrating from 0 to 1 gives
This expansion is correct to within which we can show by using the alternating series estimation theorem. We know that
is decreasing. The terms also limit to 0 and clearly this is an alternating series. Hence, the error above using three terms is no worse than what the fourth term tells us it is. Hence, the error is bounded above by
and this completes the question.