Taylor Series

Mathematical constants like and e or functions like , , are difficult to compute. Since these functions are important for many daily engineering applications, mathematicians have spent a lot of time and energy looking for better ways to compute these functions. One method is to replace a function with its Taylor series, which is, roughly speaking, an infinitely long polynomial.

A Taylor series is the sum of a sequence of terms. In contrast to arithmetic or geometric sequences, the terms of a Taylor series depend on two unknowns: some variable x and the position i in the sequence. Here is the Taylor series for the exponential function:

That is, if we wish to compute for any specific x, we replace x with the number and determine the value of the series. In other words, for a specific value of x, say, 1, the Taylor series becomes an ordinary series, that is, a sum of some sequence of numbers:

While this series is the sum of an infinitely long sequence, it actually is a number, and it often suffices to add just a few of the first few terms to have an idea what the number is.

The key to computing a Taylor series is to formulate each term in the underlying sequence as a function of x and its position i. In our running example, the Taylor sequence for the exponential function has the shape

Assuming a fixed x, here is an equivalent Scheme definition:

;; e-taylor : N -> number
(define (e-taylor i) 
  (/ (expt x i) (! i)))

;; ! : N -> number 
(define (! n) 
  (cond 
    [(= n 0) 1] 
    [else (* n (! (sub1 n)))])) 

Putting everything together, we can define a function that computes the xth power of e. Since the function requires two auxiliaries, we use a local:

(define (e-power x)
  (local ((define (e-taylor i) 
            (/ (expt x i) (! i))) 
          (define (! n) 
            (cond 
              [(= n 0) 1] 
              [else (* n (! (sub1 n)))]))) 
    (series 10 e-taylor))) 

Exercise 23.3.6

Replace 10 by 3 in the definition of e-power and evaluate (e-power 1) by hand. Show only those lines that contain new applications of e-taylor to a number.

The results of e-power are fractions with large numerators and denominators. In contrast, Scheme's built-in exp function produces an inexact number. We can turn exact fractions into inexact numbers with the following function:

;; exact->inexact : number [exact] -> number [inexact] 

Exercise 23.3.7

Develop the function ln, which computes the Taylor series for the natural logarithm. The mathematical definition of the series is

This Taylor series has a value for all x that are greater than 0.

DrScheme also provides log, a primitive for computing the natural logarithm. Compare the results for ln and log. Then use exact->inexact (see exercise ) to get results that are easier to compare. Solution

Exercise 23.3.8

Develop the function my-sin, which computes the Taylor series for sin, one of the trigonometric functions. The Taylor series is defined as follows:

It is defined for all x.

Hint: The sign of a term is positive if the index is even and negative otherwise. Mathematicians compute to determine the sign; programmers can use cond instead. Solution

Exercise 23.3.9

Mathematicians have used series to determine the value of for many centuries. Here is the first such sequence, discovered by Gregory (1638-1675):

Define the function greg, which maps a natural number to the corresponding term in this sequence. Then use series to determine approximations of the value of .

Note on : The approximation improves as we increase the number of items in the series. Unfortunately, it is not practical to compute with this definition. Solution