Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Example 8.5.2
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 8.5.2(a) where is controlled by the slider.
As , but the limit is not uniform in . As increases, the interval on which the remainder gets close to zero increases to the right.
=
Figure 8.5.2(a) Slider-controlled graph of
In the limit as , the interval on which the remainder approaches zero becomes the whole real line.
The Maclaurin series is because and for
Maple Solution
The expression for
Write Context Panel: Assign Name
Show that as
Calculus palette: Limit template Context Panel: Evaluate and Display Inline
Obtain the Maclaurin series
Write the exponential function, being sure to use a Maple template or palette for "e".
Context Panel: Series≻Formal Power Series Complete the dialog as per Figure 8.5.2(b).
Figure 8.5.2(b) Formal Power Series dialogc
Obtain the Maclaurin series from first principles
Write (Use the correct "e"!) Context Panel: Assign Function
Expression palette: Summation template
Context Panel: 2-D Math≻Convert To≻Inert Form
Context Panel: Evaluate and Display Inline
Note: The symbol for the nth-derivative can be typed, or it can be obtained as the template in the Calculus palette.
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