Math Insight

Classic examples of Taylor polynomials


Some of the most famous (and important) examples are the expansions of ${1\over 1-x}$, $e^x$, $\cos x$, $\sin x$, and $\log(1+x)$ at $0$: right from the formula, although simplifying a little, we get \begin{align*} {1\over 1-x}&=1+x+x^2+x^3+x^4+x^5+x^6+\ldots\\ e^x&=1+{x\over 1!}+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}+\ldots\\ \cos x&=1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}+{x^8\over 8!}\ldots\\ \sin x&={x\over 1!}-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}+\ldots\\ \log(1+x)&=x-{x^2\over 2}+{x^3\over 3}-{x^4\over 4}+{x^5\over 5}- {x^6\over 6}+\ldots \end{align*} where here the dots mean to continue to whatever term you want, then stop, and stick on the appropriate remainder term.

It is entirely reasonable if you can't really see that these are what you'd get, but in any case you should do the computations to verify that these are right. It's not so hard.

Note that the expansion for cosine has no odd powers of $x$ (meaning that the coefficients are zero), while the expansion for sine has no even powers of $x$ (meaning that the coefficients are zero).

At this point it is worth repeating that we are not talking about infinite sums (series) at all here, although we do allow arbitrarily large finite sums. Rather than worry over an infinite sum that we can never truly evaluate, we use the error or remainder term instead. Thus, while in other contexts the dots would mean ‘infinite sum’, that's not our concern here.

The first of these formulas you might recognize as being a geometric series, or at least a part of one. The other three patterns might be new to you. A person would want to be learn to recognize these on sight, as if by reflex!