Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
If an=n+5−n, determine if the sequence ann=0∞ converges or diverges.
If it converges, find the limit of the sequence.
In the limit as n→∞, the general term of the sequence assumes the indeterminate form ∞−∞. Table 3.9.1 suggests several ways to transform an to modify this indeterminate form, and Table 3.9.2 provides a formal algebraic modification, although a difficult one. The simplest change is rationalization, whose steps are shown below.
In this form, an is easily seen to converge to zero as n→∞.
Calculus palette: Limit operator
Context Panel: Evaluate and Display Inline
limn→∞n+5− n = 0
Maple's limit operator treats n as a continuous variable. In fact, the Maple developer who maintains the limit command has more than once informed this author that he knows of no algorithm by which a limit can be taken through a discrete variable. In most instances in calculus, this treatment of a discrete index as if it were a continuous variable, does not lead to an erroneous result. But occasionally, a sequence will be encountered where this failure of automatic calculation to provide a limit through the integers will give a misleading result. A simple example is the sequence whose general term is sinn π en. Each term of the sequence is zero, so the limit is trivially zero, but Maple will declare the limit of this sequence to be undefined (i.e., does not exist) because the continuous algorithm that is used sees numbers between −1 and 1 multiplied by larger and larger exponentials.
Table 8.1.13(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 8.1.3(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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