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Let f(x) = 1/(1-2x)
(a) Prove by induction that for k ≥ 1
f(k)(x) = 2kk! / (1-2x)k+1
(b) Show that the Taylor series for f at a=0 is given by
Σ∞k=0 = 2kxk
(c) What is the radius of convergence of the power series in (b)?
(d) Use the Taylor polynomial of degree 4 to give an approximation to f(1/10).
(e) Use the error term from Taylor’s Theorem to find an upper bound for the error in (d).
(f) Based on your result on (b) and (c), find the Taylor series for g(y) = 1/(1-y2) at a = 0 and its radius of convergence.
Solution of the example in word.