MATH101 April 2008
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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.
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The Maclaurin series for is
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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.
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