Find two different Taylor series to represent the function f(x) = 1/x such that one series is

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Find two different Taylor series to represent the function

f(x) = 1/x

such that one series is

f(x) = a0 + a1(x − 1) + a2(x − 1)2 +· · ·

and the other is

f(x) = b0 + b1(x − 2) + b3(x − 2)2 +· · ·

Show that bn = an/2n for any value of n. Find the interval of convergence for each series (the ratio test may be used). Which series must you use in the vicinity of x = 3? Why?

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