\(f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...\)
\(f(x) = \frac{1}{1-x}\)
Evaluated at a = 0:
\(f(x) = 1 + \frac{\frac{d}{dx}(\frac 1{1-x})(0)}{1!}x + \frac{\frac{d^2}{dx^2}(\frac 1{1-x})(0)}{2!}x^2 + \frac{\frac{d^3}{dx^3}(\frac 1{1-x})(0)}{3!}x^3 + \frac{\frac{d^4}{dx^4}(\frac 1{1-x})(0)}{4!}x^4\)
\(f(x) = 1 + \frac{(\frac 1{(1-x)^2})(0)}{1!}x + \frac{(\frac 2{(1-x)^3})(0)}{2!}x^2 +\frac{(\frac 6{(1-x)^4})(0)}{3!}x^3 +\frac{(\frac {24}{(1-x)^5})(0)}{4!}x^4\)
Evaluating at x = 0:
\(f(x) = 1 + \frac{1}{1!}x + \frac{2}{2!}x^2 + \frac {6}{3!}x^3 + \frac{24}{4!}x^4\)
\(\frac 2{2!} = 1, \frac 6{3!} = 1, \frac {24}{4!} = 1\)
\(f(x) = 1 + x + x^2 + x^3 + x^4 + ...\)
More generally,
\[f(x) = \sum_{n = 0}^{\infty} x^n \]
\(f(x) = e^x\)
\(f(x) = 1 + \frac{\frac{d}{dx}(e^x)(0)}{1!}x + \frac{\frac{d^2}{dx^2}(e^x)(0)}{2!}x^2 + \frac{\frac{d^3}{dx^3}(e^x)(0)}{3!}x^3 + \frac{\frac{d^4}{dx^4}(e^x)(0)}{4!}x^4\)
\(f(x) = 1 + \frac{(e^x)(0)}{1!}x + \frac{(e^x)(0)}{2!}x^2 + \frac{(e^x)(0)}{3!}x^3 + \frac{(e^x)(0)}{4!}x^4\)
Evaluating the derivative at x = 0:
\(f(x) = 1 + \frac{1}{1!}x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4\)
\(f(x) = 1 + x + \frac {x^2}{2} + \frac {x^3}{6} + \frac {x^4}{24} + ...\)
More generally,
\[f(x) = \sum_{n = 0}^{\infty} \frac {x^n}{n!} \]
\(f(x) = ln(1 + x)\)
\(f(x) = 1 + \frac{\frac{d}{dx}(ln(1 + x))(0)}{1!}x + \frac{\frac{d^2}{dx^2}(ln(1 + x))(0)}{2!}x^2 + \frac{\frac{d^3}{dx^3}(ln(1 + x))(0)}{3!}x^3 + \frac{\frac{d^4}{dx^4}(ln(1 + x))(0)}{4!}x^4\)
\(f(x) = 1 + \frac{\frac{1}{1+x}(0)}{1!}x - \frac{(\frac {1}{(1+x)^2})(0)}{2!}x^2 + \frac{\frac {2}{(1 + x)^3}(0)}{3!}x^3 - \frac{\frac{6}{(1 + x)^4}(0)}{4!}x^4\)
\(f(x) = 1 + \frac{1}{1!}x - \frac{1}{2!}x^2 + \frac{2}{3!}x^3 - \frac{6}{4!}x^4\)
\(f(x) = 1 + x - \frac {x^2}2 + \frac{x^3}3 - \frac{x^4}4\)
More generally,
\[f(x) = \sum_{n = 1}^{\infty} (-1)^{n + 1} \frac {x^n}{n} \]
\(f(x) = x^{\frac 12}\)
\(f(x) = 1 + \frac{\frac{d}{dx}(x^\frac 12)(0)}{1!}(x-a) + \frac{\frac{d^2}{dx^2}(x^\frac 12)(0)}{2!}(x-a)^2 + \frac{\frac{d^3}{dx^3}(x^\frac 12)(0)}{3!}(x-a)^3 + \frac{\frac{d^4}{dx^4}(x^\frac 12)(0)}{4!}(x-a)^4\)
\(f(x) = 1 + \frac{(\frac {1}{2 \sqrt{x}})(0)}(x-a)x - \frac{(\frac {1}{4x^{\frac 32}})(0)}{2!}(x-a)^2 + \frac{(\frac{3}{8x^{\frac52}})(0)}{3!}(x-a)^3 - \frac{(\frac{15}{16x^{\frac72}})(0)}{4!}(x-a)^4\)
Undefined at 0, so evaluate for x = 1:
\(f(x) = 1 + \frac {1}{2}(x - 1) - \frac 18 (x - 1)^2 + \frac {1}{16} (x - 1)^3 - \frac{5}{128}(x - 1)^4 + ...\)
I cannot find a simple pattern to represent the sum for this sequence.