Using the representation formula in Taylor's
Theorem for a series centered at $a$
(including at $a=0$), $$f(x) = \sum_{n=0}^\infty
\frac{f^{(n)}(a)}{n!} (x-a)^n,$$ we can derive the power series
representation for a number of other common functions. We
call these Taylor series expansions,
or Taylor series.

We will compute the Taylor series of several functions, all
centered at 0. Recall from our previous practice the
following steps: First, compute the first,
second, third, etc. derivatives of $f$ until you see a pattern and
can write $f^{(n)}(x)$ with some certainty. Next, you can compute
$f^{(n)}(0)$, and then finally, find (and simplify if
necessary) $\frac{f^{(n)}(0)}{n!}x^n$.

Or, manipulate a series you
know, i.e. if you have the Taylor series for $f(x)$ you can find
the series for $f(3x)$ without recomputing.

DO: Compute the Taylor series
for $f(x)=e^x$ and $f(x)=\sin x$.

DO: Once you have found the
Taylor series for $e^x$, find the Taylor series for $e^{2x}$
without recomputing -- just replace all the $x$-values with
$2x$.

The video computes these and other Taylor series -- try your own
before watching these solutions.