List of Maclaurin Series for 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: \((-1,1)\)
    • Radius of Convergence: 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: \(1\)