Science:Math Exam Resources/Courses/MATH101/April 2016/Question 11 (b)
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Question 11 (b)
Both parts of this question concern the series .
(b) Suppose that you approximate the series by its fifth partial sum . Give an upper bound for the error resulting from this approximation. Explain why your error bound is valid for this series.
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Recall that Alternating series estimation theorem:
If is the sum of an alternating series that satisfies:
then , where .
(Alternative way) Since , try to estimate the right hand side, using the Remainder estimate for the integral test:
Suppose where is a continuous, positive, decreasing function for and is convergent. Then,
Compare this error with the one obtained in Hint 1.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
In part (a), to show the convergence of the series we checked that is decreasing and positive. The limit of this sequence is equal to , because by applying l'Hopital Rule:
Thus by the Alternating Series Test, this series is convergent.
For a convergent Alternating Series, as mentioned in hint (1), we would like to approximate the sum with the sum of the first five terms as the following
By the error formula for Alternating Series, the upper bound for the error of approximation is the immediate term of the sequence after which is
This can be justified by the following argument:
Let's write the error term in the expanded form:
By playing with brackets we can find the bound for the error. First:
due to the decreasing terms of the sequence, the sum of the terms in each of the ( ) will be a negative number, so
Again because of decreasing terms, the sum of the terms in each ( ) is positive
so , which implies
An alternative method is to use the Integral test to approximate the error.
First we note that
so it is sufficient to find an upper bound for
Now if , then we showed that is positive and decreasing, so we can apply the integral test to estimate with .
This integral represents the area under the graph of