Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Show that the formal Maclaurin expansion of fx=e−1/x2x≠00x=0is identically zero, so that this expansion does not represent fx. Hint: Show that fn0=0 for n=1,2,….
Showing that fn0=0 for n=1,2,…, requires two things:
What are the derivatives of a piecewise-defined function at the point where the rule for the function changes?
If k is a positive integer, what is the limit of e−1/x2/xk as x→0, and how is it obtained?
If x≠0, f′x=2⁢ⅇ−1x2x3, a result obtained by differentiating the rule for f when x≠0.
To obtain f′0, apply the limit-definition of the derivative:
The actual value of the limit is obtained by applying L'Hôpital's rule to the indeterminate form 0/0, with the fraction written with 1/h in the numerator.
Thus, f′x=2e−1/x2x3x≠00x=0, a result needed to obtain f″0.
If x≠0, f″x=4x6−6x4e−1/x2, a result obtained by differentiating the rule for f′x when x≠0.
To obtain f″0, evaluate limh→02e−1/h2h3h=2limh→0e−1/h2h4. This limit also yields to L'Hôpital's rule if the numerator is taken as 1/h4, and the rule is applied multiple times.
h−4e1/h2→−4 h−5−2h3e1/h2→2h−2e1/h2→2 −2 h−3−2h3e1/h2=2 e−1/h2→0
Thus, f″x=4x6−6x4e−1/x2x≠00x=0, a result needed to obtain f‴0.
At this point, appropriate generalizations need to be made, both for the pattern of the derivatives and for the pattern of the limits. Each new derivative will require computing the limit of a fraction of the form h−k/e1/h2, which will yield to multiple applications of L'Hôpital's rule.
The Taylor-series recipe applied to fx given in this example is
∑n=0∞fn0n!xn =0+0+⋯ =0
a sum of terms all of which are zero, so that "series" is just zero. It does not represent fx even though all the derivatives of fx exist.
The code behind Maple's Limit Methods tutor proves most useful in applying L'Hôpital's rule multiple times. To make use of this code as painless as possible, it will be implemented in its command-form, thereby eliminating the tedium of launching the tutor repeatedly. Note also that the Rule command must be applied to the inert form of the limit operator. Conversion to inert is via the "2-D Math≻Convert To≻Inert Form" option in the Context Menu for the operator. The display of an inert operator is gray, rather than the default black.
Access the Student Calculus1 package
Tools≻Load Package: Student Calculus 1
Define e−1/x2 as the function gx
Context Panel: Assign Function
gx=ⅇ−1/x2→assign as functiong
Obtain f′x for x≠0
Context Panel: Evaluate and Display Inline
g′x = 2⁢ⅇ−1x2x3
Obtain f′0 by evaluating limh→0gh/h
Apply the Rule command tailored to invoke L'Hôpital's rule.
Obtain f″x for x≠0
Contest Menu: Evaluate and Display Inline
g″x = −6⁢ⅇ−1x2x4+4⁢ⅇ−1x2x6
Obtain f″0 by evaluating limh→0g′h/h
Take each term separately; each such term must be divided by h.
No need to divide by h - simply apply L'Hôpital's rule again.
An application of the Product rule for limits would show that the limits for both factors are zero, hence, the desired limit is zero.
Consider the second term: divide by h and apply L'Hôpital's rule. The result has already been considered.
These were the very calculations that were implemented in order that the Mathematical Solution would be correct. Here is an instance where Maple proves to be exceedingly helpful in establishing the pattern that has to be generalized.
An alternative to the use of the Rule command is the use of the Limit Methods tutor. Figure 8.5.1(a) shows the state of the tutor after L'Hôpital's rule is applied once to the limit limh→0gh/h5. Figure 8.5.1(b) shows how the rule is applied within the tutor.
Figure 8.5.1(a) The Limit Methods tutor
Figure 8.5.1(b) L'Hôpital's rule applied within the Limit Methods tutor
The upper portion of Figure 8.5.1(b) shows the initial state of the rule. The numerator and denominator must be made to change places. Select, in turn, the numerator and denominator, and use the arrows to change their respective positions. The result is shown in the lower portion of Figure 8.5.1(b). When the "Apply" button is pressed, the tutor assumes the state shown in Figure 8.5.1(a).
L'Hôpital's rule is to be applied twice more in this way, resulting in a limit that yields to the Product rule.
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