Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Example 8.5.3
Show that R^n+1x for fx=sinx goes to zero as n→∞, establishing that f has a Maclaurin series.
Find the terms of that series.
Solution
Mathematical Solution
The absolute value of the Taylor-series remainder for fx=sinx is
R^n+1x={sinx |x|n+1n+1!n oddcosx |x|n+1n+1!n even
For each fixed x, R^n+1x→0, as suggested by Figure 8.5.3(a) where n is controlled by the slider.
n= =
Figure 8.5.3(a) Slider-controlled graph of R^n+1
As n→∞, |R^n+1x|→0 but the limit is not uniform in x. As n increases, the interval on which the remainder gets close to zero increases to the right and left.
In the limit as n→∞, the interval on which the remainder approaches zero becomes the whole real line.
The Maclaurin series is fx=sinx=∑n=0∞−1nx2n+12n+1! because the derivatives of sinx form the repeating sequence cosx,−sinx,−cosx,sinx, whose values at x=0 are 1,0,−1,0.
The simplest way to show that the remainder term goes to zero is to note that it will alternately contain either sinx or cosx. Instead of taking the limit of such a problematic expression, estimate R^n+1 by noting that the absolute value of either trig function is bounded by 1. If something bigger than the absolute value of the remainder goes to zero, so also does the remainder itself.
Maple Solution
The estimate for R^n
Write R=… Context Panel: Assign Name
R= x2 n+12 n+1!→assign
Show that R→0 as n→∞
Calculus palette: Limit template Context Panel: Evaluate and Display Inline
limn→∞R = 0
Obtain the Maclaurin series
Write sinx
Context Panel: Series≻Formal Power Series Complete the dialog as per Figure 8.5.3(b).
Figure 8.5.3(b) Formal Power Series dialog
sinx→formal series∑n=0∞−1nx2n+12n+1!
Obtain the Maclaurin series from first principles
Write fx=… Context Panel: Assign Function
fx=sinx→assign as functionf
Expression palette: Summation template
Context Panel: 2-D Math≻Convert To≻Inert Formb
Context Panel: Evaluate and Display Inline
∑n=0∞fn0n!xn = ∑n=0∞sin12nπxnn!
Control-drag the summand.
Context Panel: Evaluate at a Point≻n=2⋅k+1
Context Panel: Simplify≻Assuming Integer
sin12nπxnn!→evaluate at pointsin122k+1πx2k+12k+1!→assuming integer−1kx2k+12k+1!
Note: The symbol for the nth-derivative can be typed, or it can be obtained as the template fnx 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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