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Control Number of Terms in Series Expansions

Taylor series expansions approximate an arbitrary expression with a polynomial. The number of terms in a series expansion determines the accuracy of the approximation. The number of terms in a series expansion depends on the truncation order of the expansion. By default, MuPAD® computes the first six terms of series expansions:

taylor(exp(x), x)

The number of terms includes the terms with coefficients equal to zero. For example, the taylor series expansion of cos(x) includes the terms 0 x, 0 x3, and 0 x5. MuPAD computes these terms, but does not display them:

taylor(cos(x), x)

Suppose, you want to approximate an exponential function with the polynomial expression around x = 0. Use the third parameter in taylor to specify the order of series expansion. For example, compute the series expansions approx1 specifying the truncation order 3. Compare the result with the series expansion computed for the default order:

exact := exp(x):
approx1 := taylor(exp(x), x, 3);
approx2 := taylor(exp(x), x)

Plot the exact expression, exact, and its taylor series expansions, approx1 and approx2, in the same coordinate system. The series expansion with more terms approximates the expression exp(x) better:

plot(
  plot::Function2d(exact, x = -PI..PI,
                   Legend = "exp(x)",
                   Color = RGB::Red),
  plot::Function2d(approx2, x = -PI..PI,
                   Legend = "approximation of exp(x), up to O(x^6)",
                   Color = RGB::Blue),
  plot::Function2d(approx1, x = -PI..PI,
                   Legend = "approximation of exp(x), up to O(x^3)",
                   Color = RGB::Green)
)

There are two ways to change the truncation order for series expansions:

  • Locally by passing the truncation order as the third parameter to taylor. By using this parameter, you specify the truncation order for a particular series expansion. All other series expansions use the default order. The parameter is available for the following commands: taylor, mtaylor, and series. For more information, see the help pages for these commands.

  • Globally by using the environment variable ORDER. When you change this variable, all series expansions use the new truncation order.

To change the truncation order for a particular series expansion, pass the new order as a third parameter to taylor:

taylor(exp(x), x, 10)

To change the default truncation order for all series expansions, modify the environment variable ORDER:

ORDER := 7:
taylor(exp(x), x)

The following computations use the new value of ORDER:

taylor(sqrt(1 - x), x)

To restore the default value of ORDER, use the delete command:

delete ORDER:
taylor(sqrt(1 - x), x)

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