“Taylor series of a real or complex-valued function f(x) that is infinitely differentiable at a real or complex number a is the power series” -wikipedia
\(f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...\)
\(f'(x) = \frac{1}{(1-x)^2}\)
\(f''(x) = \frac{2}{(1-x)^3}\)
\(f'''(x) = \frac{6}{(1-x)^4}\)
a = 0
converges for \(|x| < 1\)
\(f(0) + \frac{\frac{1}{(1-0)^2}}{1!}(x-0) + \frac{\frac{2}{(1-0)^3}}{2!}(x-0)^2 + \frac{\frac{6}{(1-0)^4}}{3!}(x-0)^3 + ...\)
\(1 + x + x^2 + x^3 + ...\)
\(f'(x) = e^x\)
\(f''(x) = e^x\)
\(f'''(x) = e^x\)
a = 0
converges for all values of x
\(f(a) + \frac{e^0}{1!}(x-0) + \frac{x^2}{2!} + \frac{e^0}{3!}(x-0)^3 + ...\)
\(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}+ ...\)
\(f'(x) = \frac{1}{(x + 1)}\)
\(f''(x) = \frac{1}{(x + 1)^2}\)
\(f'''(x) = \frac{1}{(x + 1)^3}\)
a = 0
converges for \(|x| < 1\)
\(f(0) + \frac{\frac{1}{(0 + 1)}}{1!}(x-0) + \frac{\frac{1}{(0 + 1)^2}}{2!}(x-0)^2 + \frac{\frac{1}{(0 + 1)^3}}{3!}(x-0)^3 + ...\)
\(0 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...\)
\(f'(x) = \frac{1}{2}x^{\frac{-1}{2}}\)
\(f''(x) = \frac{-1}{4}x^{\frac{-3}{2}}\)
\(f'''(x) = \frac{3}{8}x^{\frac{-5}{2}}\)
a = 1
converges for \(|x - 1| < 1\)
\(f(1) + \frac{\frac{1}{2}1^{\frac{-1}{2}}}{1!}(x-1) + \frac{\frac{-1}{4}1^{\frac{-3}{2}}}{2!}(x-1)^2 + \frac{\frac{3}{8}1^{\frac{-5}{2}}}{3!}(x-1)^3 + ...\)
\(1 + \frac{\frac{1}{2}}{1!}(x-1) + \frac{\frac{-1}{4}}{2!}(x-1)^2 + \frac{\frac{3}{8}}{3!}(x-1)^3 + ...\)
\(1 + \frac{1}{2}(x-1) + \frac{-1}{8}(x-1)^2 + \frac{1}{16}(x-1)^3 + ...\)