Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
If an=2⁢n+5n2+7⁢n+2, determine the limit of the sequence ann=0∞.
If fx=2 x+5x2+7 x+2,x∈0,∞, then an=fn so that limn→∞an=limx→∞fx.
Since fx tends to the indeterminate form ∞∞ as x→∞, Table 3.9.1 suggests the use of L'Hôpital's rule. A simpler approach is to divide numerator and denominator by x, resulting in
Consequently, this gives limn→∞an=2.
Calculus palette: Limit operator
Context Panel: Evaluate and Display Inline
limn→∞2⁢n+5n2+7⁢n+2 = 2
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 8.1.16(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 8.1.16(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.
Table 8.1.16(b) contains the initial step of the solution provided by Maple's
tutor. Internally, Maple explores the use of L'Hôpital's rule, but switches to the algebraic approach used in the Mathematical Solution, above.
Table 8.1.16(b) Initial step of the solution by the Limit Methods tutor
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