Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Example 8.5.9
For fx=x2/1−x3, use the Binomial expansion formula to obtain its Maclaurin series.
Solution
Mathematical Solution
To apply the formalism of the Binomial series, write x2/1−x3 as x2 1−x31/2, so that c=1/2 and x in the expansion is replaced by −x3. Then, the Binomial series formula gives
x2/1−x3
= x2∑n=0∞1/2n−x3n
=x21+12x3+38x6+516x9+⋯
=x2+12x5+38x8+516x11+⋯
Of course, this expansion is equivalent to the Maclaurin series for the given function.
Maple Solution
Working from the Context Panel, Maple is unable to provide a formal power series for x2⋅1−x3−1/2, but it can provide one for 1−x3−1/2.
Obtain Maple's formal power series for 1−x3−1/2
Control-drag the given function. Context Panel: Series≻Formal Power Series Set the index to n=0
1−x3−1/2→formal series∑n=0∞2n!4−nx3nn!2
Use this expression to obtain the first few terms of the fulll expansion
Control-drag the expression for the formal power series and edit ∞, the upper limit of the sum, to say, 4.
Context Panel: Evaluate Sum
x2⋅∑n=042n!4−nx3nn!2=x21+12x3+38x6+516x9+35128x12
With an application of the appropriate command, Maple does succeed in obtaining a formal power series for the given function.
convertx2−x3+1,FormalPowerSeries = ∑k=0∞2k!4−kx3k+2k!2
Obtain the first few terms of the Maclaurin series for x2/1−x3
Control-drag the given function and press the Enter key.
Context Panel: Series≻Series≻x The order of the series is the power in the remainder term. To obtain an expansion containing up to terms of 4th degree, the order has to be taken as 12. To return a polynomial, and not a series data structure containing a term indicating the order of the error, check the box for "Remove order term." See Figure 8.5.9(a).
Figure 8.5.9(a) Series dialog
x2/1−x3→series in xx2+12x5+38x8+516x11
Apply the Binomial expansion from first principles
Expression palette: Summation and binomial templates
Apply the simplify command with the assuming option. (Maple lost the ability to make this evaluation syntax-free.)
simplifyx2/∑n=0∞1/2n⋅−x3n assuming x>−1,x<1 = x2−x3+1
Maple converts the formal expression for the Binomial series to the given function, thereby demonstrating that the Binomial series, as determined by the formula in Section 8.5, actually represents the given function.
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