Taylor series are a computational tool.
You probably know that . Have you ever wondered how this kind of approximation is obtained? There are many ways to do it, but one way is to use Taylor series! Here is a plan for approximating using a series:
- Find a function which takes a nice number (like , or , or ) as an input and returns something involving (like , or , or ) as an output.
- Make sure that this function has a Taylor series which we can compute easily.
- Plug the nice number into the Taylor series: We now have an algorithm for approximating .
The idea is to use the following fact: If you recall, we found that Taylor series for arctangent already, by substituting into the geometric series to find the series for , and then integrating this series to find the series for : And so Thus Really cool!
Since is a whole number, we would need to sum terms just to be assured that you’ve found an approximation within of ! Even then, you have to be careful not to accumulate too many rounding errors when performing the computations.
Part of the reason the series above converges so slowly is that the series is not absolutely convergent. Also is the right endpoint of the interval of convergence for : It just barely makes the cut between convergent and not convergent! There are more advanced series for which converge much more quickly, for example Ramanujan’s formula: This series computes eight additional decimal places for with each term of the series. Amazing! What about other approximations? You already know one way to compute the number : As Now we can also approximate using series!
Finally, we’ll show how to use Taylor series to obtain arbitrary precision when dealing with square-roots.
Taylor series can be used like L’Hôpital’s rule on steroids when evaluating limits.
First lets see why Taylor’s series subsumes L’Hôpital’s rule: Say , and we are interested in Then using Taylor series
As long as . This is exactly L’Hôpital’s rule! Let’s use this in a L’Hôpital’s rule situation, without invoking L’Hôpital’s rule directly:
We can also use this approach to limit evaluation in cases where L’Hôpital’s rule would need to be applied multiple times.
It might not seem like Taylor series would be much help evaluating limits at infinity, since Taylor series are all about approximating a function close to some given finite point. It turns out that we can still use Taylor series to study function behavior at infinity by transforming the function:
Composing with ‘‘moves infinity to zero,’’
and we can then use Maclaurin series to study the behavior. Let’s see that in action:
Sometimes we get a series as an answer to some problem (For instance, in the next section we will find series solutions to differential equations), but we would really like a closed form expression. A closed form expression is one that can be evaluated in a finite number of steps. For example
This is not always possible, but sometimes if we are insightful we can manipulate a given series into form where we can recognize it.