Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Example 8.5.8
For fx=8+x3, use the Binomial expansion formula to obtain its Maclaurin series.
Solution
Mathematical Solution
To apply the formalism of the Binomial series, write 8+x3 as 81/31+x/81/3=21+x/81/3, so that c=1/3 and x in the expansion is replaced by x/8. Then, the Binomial series formula gives
8+x3 = 2 ∑n=0∞1/3nx8n=2+112x−1288x2+520736x3−5248832x4+⋯
Of course, this expansion is equivalent to the Maclaurin series for the given function.
Maple Solution
Obtain Maple's formal power series for 8+x3
Control-drag the given function. Context Panel: Series≻Formal Power Series Set the index to n=0
8+x3→formal series∑n=0∞2−1n8−npochhammer−13,nxnn!
Use this expression to obtain the first few terms of the 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
∑n=042−1n8−npochhammer−13,nxnn!=2+112x−1288x2+520736x3−5248832x4
The pochhammer symbol is expressed in terms of the gamma function, itself a generalization of the factorial function. The details are well beyond the scope of the introductory integral calculus course.
Obtain the first few terms of the Maclaurin series for 8+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 5. To return a polynomial, and not a series data structure containing a term indicating the order of the remainder, check the box for "Remove order term." See Figure 8.5.8(a).
Context Panel: Simplify≻Simplify
Figure 8.5.8(a) Series dialog
8+x3
8+x13
→series in x
81/3+12481/3x−157681/3x2+54147281/3x3−549766481/3x4
= simplify
2+112x−1288x2+520736x3−5248832x4
Apply the Binomial expansion from first principles
Expression palette: Summation and binomial template
Apply the simplify command along with the assuming option. (Maple lost the ability to make this evaluation syntax-free.)
simplify2∑n=0∞1/3n⋅x8n assuming x>−8,x<8 = 8+x13
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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