Question

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = 2/x, a = −3

Answer 1

\[\begin{align*}f(x)&=\frac{2}{x}\\\\
f'(x)&=-\frac{2\times1}{x^2}\\\\
f''(x)&=\frac{2\times1\times2}{x^3}\\\\
f'''(x)&=-\frac{2\times1\times2\times3}{x^4}\\\\
&\vdots\\\\
f^{(n)}(x)&=\frac{2(-1)^nn!}{x^n}
\end{align*}\]
This should get you started.