# List of Maclaurin Series of Some Common Functions

$$\displaystyle f(x) = \sum_{n=0}^{\infty} c_n x^n$$
Interval of Convergence Radius of Convergence
$$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1+ x + x^2 + x^3 + \cdots$$ $$(-1,1)$$ 1
$$\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1+ x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ $$(-\infty,\infty)$$ $$\infty$$
\displaystyle \begin{aligned} \sin x & = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{(2n+1)!} \\ & = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots \end{aligned} $$(-\infty,\infty)$$ $$\infty$$
\displaystyle \begin{aligned} \cos x & = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n}}{(2n)!} \\ & = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots \end{aligned}
$$(-\infty,\infty)$$ $$\infty$$
\displaystyle \begin{aligned} \tan^{-1} x & = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{2n+1} \\ & = x – \frac{x^3}{3} + \frac{x^5}{5} – \frac{x^7}{7} + \cdots \end{aligned}
$$(-1,1]$$ $$1$$
\displaystyle \begin{aligned} \ln(1+ x) & = \sum_{n=1}^{\infty} (-1)^{n-1}\frac{x^n}{n} \\ & = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots \end{aligned}
$$(-1,1]$$ $$1$$
\displaystyle \begin{aligned} (1+ x)^k &= \sum_{n=0}^{\infty} {k \choose n} x^n \\ &= 1 + k x + \frac{k(k-1)}{2!}x^2 + \cdots \end{aligned}
$$\scriptstyle \begin{cases} [-1,1], & \text{ if } k > -1 \text{ & } k \not \in \mathbb{Z} \\ (-\infty, \infty), & \text{ if } k>-1 \text{ & } k \in \mathbb{Z} \\ (-1,1], & \text{ if } k=-1\\ \end{cases}$$
$$1$$
$$\displaystyle f(x) = \sum_{n=0}^{\infty} c_n x^n$$
Interval of Convergence Radius of Convergence
$$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$$ $$(-1,1)$$ 1
$$\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ $$(-\infty,\infty)$$ $$\infty$$
$$\displaystyle \sin x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ $$(-\infty,\infty)$$ $$\infty$$
$$\displaystyle \cos x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n}}{(2n)!}$$
$$(-\infty,\infty)$$ $$\infty$$
$$\displaystyle \tan^{-1} x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{2n+1}$$
$$(-1,1]$$ $$1$$
$$\displaystyle \ln(1+ x) = \sum_{n=1}^{\infty} (-1)^{n-1}\frac{x^n}{n}$$
$$(-1,1]$$ $$1$$
$$(1+ x)^k = \sum_{n=0}^{\infty} {k \choose n} x^n$$
$$\scriptstyle \begin{cases} [-1,1], \; & \text{ if } k > -1 \text{ & } k \not \in \mathbb{Z} \\ (-\infty, \infty), \; & \text{ if } k>-1 \text{ & } k \in \mathbb{Z} \\ (-1,1], \; & \text{ if } k=-1\\ \end{cases}$$
$$1$$
• $$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$$

• Interval of Convergence: $$(-1,1)$$
• $$\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

• Interval of Convergence: $$(-\infty,\infty)$$
• Radius of Convergence: $$\infty$$
• $$\displaystyle \sin x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$

• Interval of Convergence: $$(-\infty,\infty)$$
• Radius of Convergence: $$\infty$$
• $$\displaystyle \cos x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n}}{(2n)!}$$

• Interval of Convergence: $$(-\infty,\infty)$$
• Radius of Convergence: $$\infty$$
• $$\displaystyle \tan^{-1} x = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{2n+1}$$

• Interval of Convergence: $$(-1,1]$$
• Radius of Convergence: $$1$$
• $$\displaystyle \ln(1+ x) = \sum_{n=1}^{\infty} (-1)^{n-1}\frac{x^n}{n}$$

• Interval of Convergence: $$(-1,1]$$
• Radius of Convergence: $$1$$
• $$\displaystyle (1+ x)^k = \sum_{n=0}^{\infty} {k \choose n} x^n$$

• Interval of Convergence:
$$\begin{cases} [-1,1], & k > -1 \text{ & } k \not \in \mathbb{Z} \\ (-\infty, \infty), & k>-1 \text{ & } k \in \mathbb{Z} \\ (-1,1], & k=-1\\ \end{cases}$$
• Radius of Convergence: $$1$$