Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
Example 8.1.7
If an=1⋅3⋅5⋅⋯⋅2 n−1nn, show that the sequence ann=1∞ is decreasing.
Solution
Mathematical Solution
To show that the given sequence is decreasing, it is sufficient to show that an/an+1≥1 for all n. Write this ratio as follows.
anan+1
= 1⋅3⋅5⋅⋯⋅2 n−1nn⋅n+1n+11⋅3⋅5⋅⋯⋅2 n+1
= n+1n⋅n+1n⋅⋯⋅n+1n⋅n+12 n+1
≥1
The final inequality results because each factor in the middle line is a fraction whose value is greater than 1.
Maple Solution
Note that the numerator of an can be written with the product notation ∏k=1n2 k−1 so that its limit can be calculated in Maple as follows.
Calculus palette: Limit template
Context Panel: Evaluate and Display Inline
limn→∞∏k=1n2 k−1nn = 0
Table 8.1.7(a) contains the task template that, given the general term of a sequence, calculates and graphs its first few members.
Tools≻Tasks≻Browse: Algebra≻Sequences
Sequences
General term
expr≔∏k=1n2 k−1nn
expr:=122n+1Γn+12πnn
Index name
n
First index value
1
Last index value
10
Members
seqexpr,=..
1,34,59,105256,189625,3851728,19305117649,202702516777216,4254254782969,26189163400000000
Graph
plotseq,expr,=.., style=point, symbol=solidcircle, color=red
Table 8.1.7(a) The Sequences task template
Notice that Maple immediately represents the product in the numerator of an in terms of the gamma function because that numerator can actually be represented by factorials in the form 2 n−1!2n−1n−1!.
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