### 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!

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