Concept preview

What you will be able to do

1. 01Write the Taylor series of a function $f$ about a centre $a$ as $\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$, and specialise it to the Maclaurin series (the case $a=0$).
2. 02Recall the standard Maclaurin series for $e^x$, $\sin x$, $\cos x$, $\dfrac{1}{1-x}$, and $\ln(1+x)$, including a correct general term with its factorial and sign.
3. 03Build the series for a new function by substitution, term-by-term differentiation, or term-by-term integration of a known standard series, rather than differentiating from scratch.

If $f$ has derivatives of every order at a centre $a$, its Taylor series about $a$ is $f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^2 + \dfrac{f'''(a)}{3!}(x-a)^3 + \cdots.$ Each coefficient is a derivative of $f$ evaluated at $a$, divided by a factorial. The special case $a=0$ is the Maclaurin series, $f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)}{n!}x^n$. So "Maclaurin" is nothing more than "Taylor centred at zero" — the same formula with $a$ set to $0$.