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Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Example 8.5.5
Show that for goes to zero as , establishing that has a Maclaurin series.
Find the terms of that series.
Solution
Mathematical Solution
The Taylor-series remainder for is
For each fixed , , as suggested by Figure 8.5.5(a) where is controlled by the slider.
=
Figure 8.5.5(a) Slider-controlled graph of
As , but the limit is not uniform in . As increases, the interval on which the remainder gets close to zero increases to the right and left.
In the limit as , the interval on which the remainder approaches zero becomes the whole real line.
Maple Solution
The estimate for
Write Context Panel: Assign Name
Show that as
Calculus palette: Limit template Context Panel: Evaluate and Display Inline
Obtain the Maclaurin series
Write
Context Panel: Series≻Formal Power Series Complete the dialog as per Figure 8.5.5(b).
Figure 8.5.5(b) Formal Power Series dialog
Obtain the Maclaurin series from first principles
Write Context Panel: Assign Function
Expression palette: Summation template
Context Panel: 2-D Math≻Convert To≻Inert Formb
Context Panel: Evaluate and Display Inline
Control-drag the summand and press the Enter key.
Context Panel: Evaluate at a Point≻
Context Panel: Expand≻Expand
Context Panel: Simplify≻Assuming Integer
Note: The symbol for the nth-derivative can be typed, or it can be obtained as the template in the Calculus palette.
Note also the alternate form Maple provides for the Maclaurin series, and the additional steps it takes to transform the summand into the form obtained earlier from the Series option in the Context Panel.
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