Science:Math Exam Resources/Courses/MATH100/December 2016/Question 13 (b)/Solution 1

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By differentiating and noticing the pattern, we see that for all , .

From the mean-value form of the remainder from Taylor's theorem, we know that there exists some between 1 and 1.1 such that

If is an underestimate, then , so ; since , the condition is equivalent with

This occurs precisely when is odd; that is, when is even.

So, when is even, is an underestimate for ; in other words, when