\(\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x - c)^n\)
\(\sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!}(x - 1)^n\)
For n = 0:
\(f(1) = \frac{1^{-1}}{0!}(x - 1)^0 = 1\)
For n = 1:
\({f}^{'}(x) = -x^{-2}\)
\(f(1) = \frac{-1^{-2}}{1!}(x - 1)^1 = -x + 1\)
For n = 2:
\({f}^{''}(x) = 2x^{-3} . . . f^{''}(1) = 2\)
\(f(1) = \frac{2}{2!}(x - 1)^2 = (x - 1)^2\)
For n = 3:
\({f}^{'''}(x) = -6x^{-4} . . . f^{'''}(1) = -6\)
\(\frac{-6}{3!}(x - 1)^3 = -(x - 1)^3\)
A pattern emerges from these first few values. The sign alternates from + to -. The exponent for (x - 1) continues to increase by 1. The formula for this series is the following:
\((-1)^{n} (x - 1)^n\)