Find two different Taylor series to represent the function f(x) 1 x such that one series is f(x) a 0 a 1 (x 1) a 2 (x 1) 2 and the other is f(x) b 0 b 1 (x 2) b 3 (x 2) 2 Show that b n a n 2 n 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

Question: Find two different Taylor series to represent the function f(x) = 1/x such that one series is f(x) = a 0 + a 1 (x

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 b_n = a_n/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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