Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
If an=n!nn, find the limit of the sequence ann=1∞.
The discrete function n! is extended to the continuous variable x by the gamma function Γx, which has the property that n! =Γn+1 for integers n. Although this function is known to Maple by the name GAMMA, this path is not fruitful for the calculus student. The appropriate approach is to examine the behavior of an and generalize its behavior from the following observations.
where the fraction in parentheses, obtained by factoring out 1/n, is less than 1. Hence, by an application of Theorem 8.1.3 (the discrete form of the Squeeze theorem), 0≤limn→∞an≤limn→∞1/n=0, so the limit of the given sequence is zero.
Calculus palette: Limit operator
Context Panel: Evaluate and Display Inline
limn→∞n!nn = 0
Table 8.1.4(a) contains the
task template that, given the general term of a sequence, calculates and graphs its first few members.
First index value
Last index value
plotseq,expr,=.., style=point, symbol=solidcircle, color=red
Table 8.1.4(a) The Sequences task template
Place the cursor somewhere in the cell containing the phrase "General term"and press the Tab key often enough for the cursor to move to, and select the default general term. With this expression auto-selected, simply overwrite with the desired general term, most easily obtained by a copy/paste operation. Then, adjust any of the inputs as needed, and simply press the Enter key to execute each command in the template.
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