Suppose
is a continuous and infinitely differentiable function.
Let
for some
that is ``small''. Then
the following is true:

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This sum will always converge to the function value (for smooth functions and small enough ) if carried out to a high enough degree. Note well that the Taylor series can be rearranged to become the

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where the latter symbols stands for ``terms of order or smaller'' and vanishes in the limit. It can similarly be rearranged to form formal definitions for the second or higher order derivatives of a function, which turns out to be very useful in computational mathematics and physics.

We will find many uses for the Taylor series as we learn physics,
because we will *frequently* be interested in the value of a
function ``near'' some known value, or in the limit of very large or
very small arguments. Note well that the Taylor series expansion for
any polynomial *is* that polynomial, possibly re-expressed around
the new ``origin'' represented by
.

To this end we will find it *very* convenient to define the
following *binomial expansion*. Suppose we have a function that can
be written in the form:

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where can be

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where . is now a suitable ``small parameter'' and we can expand this expression around :

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Evaluate the derivatives of a Taylor series around to verify this expansion. Similarly, if were the larger we could factor out the and expand in powers of as our small parameter around . In that case we'd get:

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Remember,
is *arbitrary* in this expression but you should also
verify that if
is any positive integer, the series terminates and
you recover
*exactly*. In this case the ``small''
requirement is no longer necessary.

We summarize both of these forms of the expansion by the part in the brackets. Let and be an arbitrary real or complex number (although in this class we will use only real). Then:

(59) |

This is the binomial expansion, and is