Economizing a Power Series

• We begin a search for better power series representations of functions by using Chebyshev polynomials to economize a Maclaurin series.
• This will give a modification of the Maclaurin series that produces a fifth-degree polynomial
• whose errors are only slightly greater than those of a sixth-degree Maclaurin series.

• We start with a Maclaurin series for $e^x$:
  
  $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\ldots$$
  
  

• If we would like to use a truncated series to approximate $e^x$ on the interval $[0, 1]$ with a precision of $0.001$,
• We will have to retain terms through that in $x^6$, because the error after the term in $x^5$ will be more than
  
  $$1/720=0.00139~ (and~ 1/120=0.0084)$$
  
  

• Suppose we subtract
  
  $$\left(\frac{1}{720}\right)\left(\frac{T_6}{32}\right)$$
  
  
  from the truncated series.

• This will exactly cancel the $x^6$ term from Eqn. 1
• and at the same time make adjustments in other coefficients of the Maclaurin series.
• Because the maximum value of $T_6$ on the interval $[0, 1]$ is unity,
• this will change the sum of the truncated series by only
  
  $$\left(\frac{1}{720}\right)\left(\frac{1}{32}\right)<0.00005$$
  
  
  which is small with respect to our required precision of $0.001$.

• Performing the calculations, we have
  
  $$\begin{array}{ll} e^x \approx & \overbrace{1+x+\frac{x^2}{2}... ...499219x^2+\frac{x^3}{6}+0.043750x^4+\frac{x^5}{120} \end{array}$$
  
  

• The resulting fifth-degree polynomial approximates $e^x$ on $[0, 1]$ nearly as well as the sixth-degree Maclaurin series.
• its maximum error (at $x = 1$) is $0.000270$, compared to $0.000226$ for the Maclaurin polynomial (see Table 1).

Table 1: Comparison of economized series with Maclaurin series.

• We economize in that we get about the same precision with a lower-degree polynomial.
• By subtracting $\frac{1}{120}\frac{T_5}{16}$ we can economize further, getting a fourth-degree polynomial that is almost as good as the economized fifth-degree one.
• So that we have found a fourth-degree power series that meets an error criterion that requires us to use two additional terms of the original Maclaurin series.

• Because of the relative ease with which they can be developed, such economized power series are frequently used for approximations to functions.
• Much more efficient than power series of the same degree obtained by truncating a Taylor or Maclaurin series.
• Observe that even the economized polynomial of degree-4 is more accurate than a fifth-degree Maclaurin series.
• Also notice that near $x=0$, the economized polynomials are less accurate.

• We can get the economized series with MATLAB by employing our M-file for the Chebyshev series.
• We must start with $x$ as a symbolic variable, then get the Maclaurin series and subtract the proper multiple of the Chebyshev series: