Several important Maclaurin series expansions follow. We omit the derivation here but they all could be derived by utilizing the Taylor formula with remainder and showing that the remainder goes to zero.

All these expansions are valid for complex variables.

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots

for all

x \in ℝ

\frac{1 - x^{m + 1}}{1 - x} = \sum_{n = 0}^{m} x^{n}

for

x \neq 1

and

m \in ℕ_{0}

\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n}

for

| x | < 1

log (1 - x) = - \sum_{n = 1}^{\infty} \frac{x^{n}}{n}

for

| x | \leq 1, x \neq 1

\frac{x^{m}}{1 - x} = \sum_{n = m}^{\infty} x^{n}

for

| x | < 1

and

m \in ℕ_{0}

\frac{x}{{(1 - x)}^{2}} = \sum_{n = 1}^{\infty} n x^{n}

for

| x | < 1

sin x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots

for all

- \infty < x < \infty

cos x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2 n!} x^{2 n} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots

for all

- \infty < x < \infty

Refresher - List of Taylor Series of Some Common Functions