Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
If a1=1 and an+1=43+2 an for n>1, graph an for n=1,…,10.
Use Maple to find an explicit representation for an.
Use Maple to calculate limn→∞an.
The sequence an is defined by a nonlinear recurrence equation. One way to generate members of this sequence is illustrated in Table 8.1.9(a).
interfacertablesize=25:f1≔1: for k from 1 to 12 do fk+1≔4/3+2 fk; r∥k+1≔ak+1,fk+1,evalffk+1; end do: Matrix([a1,1,1,r∥2..12])
Table 8.1.9(a) Termwise generation of members of a sequence defined by a nonlinear recursion
Table 8.1.9(b) shows how to use Maple's rsolve command to obtain an explicit solution of the given recursion.
Write the recursion equation
Using the appropriate syntax, write the recursion equation.
Press the Enter key.
q≔an+1 3+2 an=4
Apply the rsolve command, using the syntax shown, and assigning the solution to the name A
Calculus palette: Limit template
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻10 (digits)
limn→∞A = 13⁢41+4110⁢41+82→at 10 digits0.8507810593
Table 8.1.9(b) Use of the rsolve command to obtain the explicit solution for an
The astute observer will notice several key points in these computations:
The loop in Table 8.1.9(a) assigns to fn, not an, anticipating the use of an in Table 8.1.9(b), and allowing the display in Table 8.1.9(a) to show the unevaluated symbols an.
The interface command in Table 8.1.9(a) raises from the default 10 to 25, the number of rows and columns a matrix can display.
The form of the recursion in Table 8.1.9(b) is essential; without changing the recursion to that form, Maple can't solve the equation.
A modest decrease in the size of the general solution can be obtained by applying the factor command to A. Although this could be done interactively via the Context Panel option Factor, the output of the command is shown in Table 8.1.9(c).
Table 8.1.9(c) Application of the factor command to the solution A
Finally, to give some credence to the claim that A is the solution of the recursion, the first few members of the sequence an are extracted from A in Table 8.1.9(d) via the seq command. Of course, this can also be done interactively via the Context Panel option Sequence.
Table 8.1.9(d) Members of the sequence an extracted from the explicit solution A
These fractions agree with those displayed in Table 8.1.9(a).
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