Finding the limit of a sequence a quotient of two power functions

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Homework Help Overview

The discussion revolves around finding the limit of a sequence defined by the quotient of two power functions as n approaches infinity. The specific expression under consideration is lim x-> infinity (3n + 2) / (5n).

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore simplifications of the expression, with suggestions to rewrite it in different forms. There are discussions about the convergence of a related geometric series and clarifications regarding the nature of the problem as a sequence rather than a series.

Discussion Status

The discussion is active, with participants providing insights into simplification techniques and questioning assumptions about the nature of the expression. Some guidance has been offered regarding the limit behavior of the sequence, although there is no explicit consensus on the approach to take.

Contextual Notes

There is mention of a homework template that may have been partially completed, indicating that certain expected elements might be missing from the original post. Participants also note the importance of correctly identifying the problem as a sequence rather than a series.

Aerospace93
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Homework Statement


lim x-> infinity 3n+2/5n
 
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I think you missed part 3 of the homework template: What did you do so far?
There is a nice way to simplify the expression.
 
can it be simplified to 5^n-2 (3/5)^n+2
 
If you add some brackets, right. There is an easier way to write it, but this one is fine as well.
You should be able to see the limit of that expression.
 
1/3^2 * (3/5)^n. the geometric series will be convergent since |r|=3/5<1?
 
3^2, not 1/3^2

Right.
 
Aerospace93 said:
1/3^2 * (3/5)^n. the geometric series will be convergent since |r|=3/5<1?
It's not a geometric series, is it? The problem statement indicates a sequence, not a series. The conclusion is correct, though: ##(3/5)^n \rightarrow 0## because ##|3/5| < 1##. (Do you know how to prove that?)
 

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