We are not going to be doing a whole lot with Taylor series once we get
out of the review, but they are a nice way to get us back into the swing of
dealing with power series. By time most
students reach this stage in their mathematical career they’ve not had to deal with
power series for at least a semester or two.
Remembering how Taylor
series work will be a very convenient way to get comfortable with power series
before we start looking at differential equations.
Taylor Series
If f(x) is an
infinitely differentiable function then the Taylor Series of f(x) about x=x0
is,
Recall that
Let’s take a look at an example.
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Example 1 Determine
the Taylor
series for  about x=0.
Solution
This is probably one of the easiest functions to find the Taylor series for. We just need to recall that,
and so we get,
The Taylor
series for this example is then,
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Of course, it’s often easier to find the Taylor series about x=0 but we don’t always do that.
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Example 2 Determine
the Taylor
series for about x=-4.
Solution
This problem is virtually identical to the previous
problem. In this case we just need to
notice that,
The Taylor
series for this example is then,
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Let’s now do a Taylor
series that requires a little more work.
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Example 3 Determine
the Taylor
series for about x=0.
Solution
This time there is no formula that will give us the
derivative for each n so let’s
start taking derivatives and plugging in x=0.
Once we reach this point it’s fairly clear that there is a
pattern emerging here. Just what this pattern
is has yet to be determined, but it does seem fairly clear that a pattern
does exist.
Let’s plug what we’ve got into the formula for the Taylor series and see
what we get.
So, every other term is zero.
We would like to write this in terms of a series, however
finding a formula that is zero every other term and gives the correct answer
for those that aren’t zero would be unnecessarily complicated. So, let’s rewrite what we’ve got above and
while were at it renumber the terms as follows,
With this “renumbering” we can fairly easily get a formula
for the Taylor
series of the cosine function about x=0.
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For practice you might want to see if you can verify that
the Taylor
series for the sine function about x=0
is,
We need to look at one more example of a Taylor series. This example is both tricky and very easy.
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Example 4 Determine
the Taylor
series for about x=2.
Solution
There’s not much to do here except to take some
derivatives and evaluate at the point.
So, in this case the derivatives will all be zero after a
certain order. That happens
occasionally and will make our work easier. Setting up the Taylor series then gives,
In this case the Taylor series terminates and only had
three terms. Note that since we are
after the Taylor
series we do not multiply the 4 through on the second term or square out the third
term. All the terms with the exception
of the constant should contain an x-2.
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Note in this last example that if we were to multiply the Taylor series we would get
our original polynomial. This should not
be too surprising as both are polynomials and they should be equal.
We now need a quick definition that will make more sense to
give here rather than in the next section where we actually need it since it
deals with Taylor
series.
Definition
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A function, f(x),
is called analytic at x=a if the Taylor series for f(x) about x=a has a positive radius of convergence and converges to f(x).
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We need to give one final note before proceeding into the
next section. We started this section
out by saying that we weren’t going to be doing much with Taylor series after this section. While that is correct it is only correct
because we are going to be keeping the problems fairly simple. For more complicated problems we would also be
using quite a few Taylor
series.