Taylor and Maclaurin Series

Definition of Taylor series:

\( f(x) = f(a) + f'(a)(x-a) + \frac{f”(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n \)

\( R_n = \frac{f^{(n)}(\xi)(x-a)^n}{n!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Lagrangue’s form )} \)

\( R_n = \frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Cauch’s form )} \)

This result holds if f(x) has continuous derivatives of order n at last. If \(\lim_{n \to +\infty}R_n = 0\), the infinite series obtained is called Taylor series for f(x) about x=a. If a=0 the series is often called a Maclaurin series.

Binomial series

\(
\begin{aligned}
(a + x)^n &= a^n + na^{n-1} + \frac{n(n-1)}{2!} a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots \\
&= a^n + { n \choose 1} a^{n-1}x + { n \choose 2} a^{n-2}x^2 + { n \choose 3} a^{n-3}x^3 + \cdots
\end{aligned}
\)

Special cases of binomial series

\( (1 + x)^{-1} = 1 – x + x^2 -x^3 + \cdots \quad -1 < x < 1 \)

\( (1 + x)^{-2} = 1 – 2x + 3x^2 – 4x^3 + \cdots \quad -1 < x < 1 \)

\( (1 + x)^{-3} = 1 – 3x + 6x^2 – 10x^3 + \cdots \quad -1 < x < 1 \)

\(
(1 + x)^{-1/2} = 1 – \frac{1}{2}x + \frac{1\cdot 3}{2\cdot 4}x^2 -
\frac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1
\)

\(
(1 + x)^{1/2} = 1 + \frac{1}{2}x – \frac{1}{2\cdot 4}x^2 +
\frac{1\cdot 3}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1
\)

Series for exponential and logarithmic functions

\( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

\( e^x = 1 + x\,\ln a + \frac{(x\,\ln a)^2}{2!} + \frac{(x\,\ln a)^3}{3!} + \cdots \)

\( \ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots \quad -1 < x \leq 1 \)

\(
\ln(1+x) = \left(\frac{x-1}{x}\right) + \frac{1}{2}\left(\frac{x-1}{x}\right)^2
+ \frac{1}{3}\left(\frac{x-1}{x}\right)^3 + \cdots \quad x \geq \frac{1}{2}
\)

Series for trigonometric functions

\( \sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots \)

\( \cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots \)

\(
\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots + \frac{2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}
\)

\(
\cot x = \frac{1}{x} – \frac{x}{3} – \frac{x^3}{45} – \cdots – \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi
\)

\(
\sec x = 1+ \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots + \frac{E_n x^{2n}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}
\)

\(
\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots + \frac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi
\)

Series for inverse trigonometric functions

\(
\arcsin x = x + \frac{1}{2}\frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7} + \cdots \quad -1 < x <1
\)

\(
\arccos x = \frac{\pi}{2} – \arcsin x =
\frac{\pi}{2} – \left(x + \frac{1}{2}\frac{x^3}{3} + \frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots \right)
\quad -1 < x < 1
\)

\(
\arctan x = \left\{
\begin{aligned}
x – \frac{x^3}{3} + \frac{x^5}{5} – \frac{x^7}{7} + \cdots & \quad -1 < x < 1 \\
\frac{\pi}{2} – \frac{1}{x} + \frac{1}{3x^3} – \frac{1}{5x^5} + \cdots & \quad x \geq 1 \\
-\frac{\pi}{2} – \frac{1}{x} + \frac{1}{3x^3} – \frac{1}{5x^5} + \cdots & \quad x < 1
\end{aligned} \right.
\)

\(
\mathrm{arccot}\,x = \frac{\pi}{2} – \arctan x = \left\{
\begin{aligned}
\frac{\pi}{2} – \left(x – \frac{x^3}{3} + \frac{x^5}{5} + \cdots \right) & \quad -1 < x < 1 \\
\frac{1}{x} – \frac{1}{3x^3} + \frac{1}{5x^5} – \cdots & \quad x \geq 1 \\
\pi + \frac{1}{x} – \frac{1}{3x^3} + \frac{1}{5x^5} – \cdots & \quad x < 1
\end{aligned} \right.
\)

Series for hyperbolic functions

\( \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \)

\( \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots \)

\(
\tanh x = x – \frac{x^3}{3} + \frac{2x^5}{15} + \cdots
\frac{(-1)^{n-1} 2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} + \cdots \quad |x| < \frac{\pi}{2}
\)

\(
\coth x = \frac{1}{x} + \frac{x}{3} – \frac{x^3}{45} + \cdots
\frac{(-1)^{n-1} 2^{2n}B_nx^{2n-1}}{(2n)!} + \cdots \quad 0 < |x| < \pi
\)

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