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Concept explainers
In this exercise we will use the Remainder Estimation Theorem to determine the number of terms that are required in Formula (14) to approximate ln2 to five decimal-place accuracy. For this purpose let
f(x)=ln1+x1−x=ln(1+x)−ln(1−x) (−1<x<1)
(a) Show that f(n+1)(x)=n![(−1)n(1+x)n+1+1(1−x)n+1]
(b) Use the result in part (a) to show that |f(n+1)(x)|≤n!⌈1(1+x)n+1+1(1−x)n+1⌉
(c) Since we want to achieve five decimal-place accuracy, our goal is to choose n so that the absolute value of the nth remainder at x=13 does not exceed the value 0.000005=0.5×10−5; that is, |Rn(13)|≤0.000005. Use the Remainder Estimation Theorem to show that this condition will be satisfied if n is chosen so that M(n+1)!(13)n+1≤0.000005 where |f(n+1)(x)|≤M on the interval [0,13].
(d) Use the result in part (b) to show that M can be taken as M=n![1+1(23)n+1]
(e) Use the results in parts (c) and (d) to show that five decimal-place accuracy will be achieved if n satisfies 1n+1[(13)n+1+(12)n+1]≤0.000005 and then show that the smallest value of n that satisfies this condition is n=13.
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Calculus Early Transcendentals, Binder Ready Version
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