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# Class Contents

## Taylor Series

### General Power Series

All of the power series that we have studied so far have been of the form
### Higher Order Derivatives

In Chapter 4 we discussed
functions which have a derivative at every point an interval
`(a,b)`. If `f(x)` is such a function, it is said to be
differentiable on `(a,b)` and its derivative is `f'`.
### Differentiating Power Series

References: Theorem 7.9 |
Lemma 7.12
`f` can be expressed as a power series in terms of its derivatives. After all, if we know all the derivatives of a function we already have to know a whole lot about it. There are however some very nice payoffs from the observations of the last few paragraphs.
### Power Series are Unique

A function cannot be expressed as a power series in more than one way. Put more precisely, if

and we showed that they converge inside disks centered on zero, and diverge outside those disks. However, we can easily do a change of variable, to study power series of the form

where `z _{0}` is fixed. This series converges on a set of the form

which is a disk of radius `R` centered on `z _{0}`.

In this section we shall be interested in real-valued functions of a real variable which come from power series. The power series will be of the form

where `x` and `x _{0}` are real numbers. The region in the
real line where this series converges is the interval

Now it
may be
that the function `f'` is also differentiable on `(a,b)`
and, if so, then its derivative, `f''` is called the **second
derivative** of `f`. By the same token, if `f''` is
differentiable on `(a,b)` then its derivative is called the third
derivative of `f` on `(a,b)`. In general, if we can repeat
this process `k` times then we say that `f` is
`k`-times differentiable, and that the last function obtained in
this way by repeated differentiation is the `k`th derivative of
`f`.

**Warning:**

Just as there is no reason why an arbitrary function
should be differentiable at all, so there is no reason, just because a
function can be differentiated `k` times, to assume that it can
be differentiated `k+1` times!

**Notation:**

We write

for the ** kth derivative of f**, if it exists. By
convention we say

If

is a power series, then recall that `f(x _{0})=a_{0}`. In other words,
the value of the power series at

Conversely,

Thus,

We have shown that the values of the derivatives of `f` at just
one point (namely `x _{0}`) determine the values of

for all `x` in a non-empty open interval, then `a _{n}=b_{n}` for all

Question: Can you see why this is?