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## Taylor approximations; the error term; convergence

The -th Taylor approximation based at to a function is the -th partial sum of the Taylor series:

Note that is a sum of terms and is a polynomial of degree at most in .
Then has the characteristic property that its derivatives agree with those of the function , when both are evaluated at , up to and including the -th derivative.

Consider now the difference .
Intuitively this should be small.
There are various estimates of the size of this difference: one is the following:

This estimate is valid throughout the interval , for a fixed positive , where the quantity is the maximum of on that interval.
So for example, for the function , we have and

For the functions and , we know that is a value of one of the two functions or , somewhere on the interval , which can never be larger than , so we always have the following estimate:

For each of these functions, we notice that as , the error goes to zero, since the denominator grows much faster than any power of the form for fixed .

When the error goes to zero as goes to infinity, we get two by-products:
• First the Taylor series converges on .
• Second the Taylor series actually represents the function on the interval .
So we can conclude as stated earlier, that the Taylor series for the functions , and always represents the function, on any interval , for any reals and , with .
Since this is true for any real , these Taylor series represent the functions on the entire real line.

As another example consider the function and its expansion based at 0.
We have , so, on the interval , where , we get and then we have:

This goes to zero as , provided .
Note that must be restricted to the range , since the function and its derivatives blow up as .
We conclude that the Taylor series represents the function on the interval , for any , so therefore also on the interval .

Finally, if a Taylor series converges on an open interval , then it converges absolutely on that interval.

Next: Tricks with Taylor series Up: 23014convergence Previous: Taylor series based at
George A. J. Sparling 2003-12-08