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Let f : R \ {1} → R be given by f(x) = 1 1 −...

Let f : R \ {1} → R be given by f(x) = 1 1 − x . (a) Prove by induction that f (n) (x) = n! (1 − x) n for all n ∈ N. Note: f (n) (x) denotes the n th derivative of f. You may use the usual differentiation rules without further proof. (b) Compute the Taylor series of f about x = 0. (You must provide justification by relating this specific Taylor series to general Taylor series.) (c) Determine the interval of convergence of the Taylor series for f(x). Justify your conclusions with appropriate convergence tests or facts from this subject. (d) At which point(s) x ∈ R does the Taylor series converge absolutely? At which point(s) does it converge conditionally? Justify your answer. (e) Using your result in part (a) (or otherwise) use a Taylor series to show that log 2 > 11/12.

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Answer #1

You can check that log(2) is not grater than (11/12) by using a calculator! It will be (7/12).

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