Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
If an=1⋅3⋅5⋅⋯⋅2 n−1nn, show that the sequence ann=1∞ is decreasing.
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.
= 1⋅3⋅5⋅⋯⋅2 n−1nn⋅n+1n+11⋅3⋅5⋅⋯⋅2 n+1
= n+1n⋅n+1n⋅⋯⋅n+1n⋅n+12 n+1
The final inequality results because each factor in the middle line is a fraction whose value is greater than 1.
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.
First index value
Last index value
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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