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

[Aerospace93]

Homework Statement

lim x-> infinity 3ⁿ⁺²/5ⁿ

 

[mfb]

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.

 

[Aerospace93]

can it be simplified to 5^n-2 (3/5)^n+2

 

[mfb]

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.

 

[Aerospace93]

1/3^2 * (3/5)^n. the geometric series will be convergent since |r|=3/5<1?

 

[mfb]

3^2, not 1/3^2

Right.

 

[jbunniii]

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?)