Maclaurin Series of Familiar Functions

e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + ⋯ = ∑_{j} \frac{x^{j}}{j!}

sin x = x − \frac{x^{3}}{3!} + \frac{x^{5}}{5!} − ⋯

cos x = 1 − \frac{x^{2}}{2!} + \frac{x^{4}}{4!} − ⋯

\frac{1}{1 − x} = (1 − x)^{−1} = 1 + x + x^{2} + ⋯ = ∑_{j} x^{j} (“geometric series”)

ln(1 − x) = −x − \frac{x^{2}}{2} − \frac{x^{3}}{3} − ⋯ = − ∑_{j} \frac{x^{j}}{j}

(corresponding to the Taylor series ln x = (x−1)− \frac{1}{2}(x−1)^{2}+ \frac{1}{3}(x−1)^{3}−⋯)

\sqrt{1 + x} = (1 + x)^{\frac{1}{2}} = 1 + \frac{x}{2} − \frac{x^{2}}{8} + ⋯

(corresponding to the Taylor series \sqrt{x} = 1 + \frac{1}{2}(x − 1) − \frac{1}{8}(x − 1)^{2} + ⋯)

(1 + x)^{− \frac{1}{2}} = 1 − \frac{x}{2} + \frac{3}{8} x^{2} − ⋯

And in general,

(1 + x)^{p} = 1 + px + \frac{p(p − 1)}{2!} x^{2} + ⋯ (“binomial series”) which becomes a familiar finite polynomial when p is a positive integer: for example,

(1 + x)^{3} = 1 + 3x + 3x^{2} + x^{3} (exactly)