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Now let us have a look at that strange integral in the end of the series expansion we got in the end of the previous page.

We call the integral the error term or rest term written *R _{k}*(

*x*). Say we have that, in the interval from 0 to

*x*, we have

for some values *N* and *M*, and that the (*k*+1)^{th} derivative is continuous. That means that

or in other words that

This means that there must exist a value between *N* and *M* , say *P*, such that that

and since the function where continuous, and varying between *N* and *M*, then there must exist a value c between 0 and *x* such that

The above might be best illustrated by a figure:

This is actually an application of the so called mean value theorem.

Putting it together we have that

for some value *c* in 0 to *x*. This is called the Lagrange form of the error term.

**An upper bound of the error**

We mostly use this find an upper bound (a value we will not exceed) for the error. What we do is to find the maximum value of the (k+1)^{th} derivative in the interval (i.e., *M*) or at least some value that we know will be bigger than the maximum. Then we know that

**An example**

Say we want to use the series expansion

The rest term will now be

Say we will use 10 terms, and *x* is between -2 and 2, what would the maximum error not exceed? The exponential function is biggest at the upper end of the interval, and so is *x*^{11}. So we get

When comparing this to the actual error we find that the actual error is about 6 times smaller. The above will thus give use a guaranteed upper bound of the error, but the error is usually much smaller.

So how many terms do we need to have an error not exceeding say 10^{-10} when *x* is between –1 and 1? We get that

or

With a bit of testing we find that *k*=13.

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