Let’s say you need to find the square root of 4.7. You pull out your hand calculator (or the Calculator app on your phone), type in , and you have the answer.
But how did the calculator know it?
In fact, we take modern computational conveniences for granted. The hand calculator (or the corresponding app) is a fantastic product of engineering; and in this exercise, we’re going to apply one of the techniques they use to calculate square roots: Taylor Series.
Taylor series allow us to write any differentiable function f(x) as an infinite series of the form
for values of x near x = a. Note that represents the n’th derivative of the function f(x) evaluated at x = a.
Clearly, the details of a Taylor series expansion depend upon the function involved; for the square root function , it can be shown that:
- The Taylor series for will be an alternating series; that is, the signs of consecutive terms will alternate from positive to negative and back again, with the pattern repeating indefinitely.
- If we assume , then the absolute value of each term is smaller than the one before.
Combined, these two facts tell us that the Taylor series for converges; and if we truncate the series after n terms, then the error in our approximation will be smaller than the absolute value of term n+1 in the series.
With this background, here is your assignment:
- Determine the number of terms in the corresponding Taylor series expansion required to approximate the value of to within , and state the resulting approximate value of .
- Use the absolute value of the first term you omitted to estimate the error in your approximation.
Use this table to organize your work:
|Function and derivatives||Evaluate function and derivatives||term of Taylor Series||term of Tayler Series evaluated at value of interest within||Cumulative sum of Taylor Series terms||Approximation accurate to within||Error estimate|