Proving convergence of a series without knowing the sum

[student93]

Homework Statement

Problem is attached in this post.

Homework Equations

Problem is attached in this post.

The Attempt at a Solution

I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm supposed to tell as to whether or not this is a geometric series.

 

[Dick]

Homework Statement

Problem is attached in this post.

Homework Equations

Problem is attached in this post.

The Attempt at a Solution

I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm supposed to tell as to whether or not this is a geometric series.

It's not geometric because the ratio of consecutive terms isn't a constant, but that ratio does approach a limit. Is that enough of a clue?

 

[student93]

It's not geometric because the ratio of consecutive terms isn't a constant, but that ratio does approach a limit. Is that enough of a clue?

Would the limit have to be 0 for the series to convergent?

Lim (3^n + 4^n)/(7^n) n -> ∞ = 0 For the series to be convergent? (I actually tried this, but can't seem to find a method to solve for such a limit).

 

[Dick]

Would the limit have to be 0 for the series to convergent?

Lim (3^x + 4^x)/(7^x) n -> ∞ = 0 For the series to be convergent? (I actually tried this, but can't seem to find a method to solve for such a limit).

You mean Lim (3^n + 4^n)/(7^n) n -> ∞ = 0. You should be able to. Divide numerator and denominator by 7^n. But that's not what I'm talking about. I'm talking about the ratio $\frac{a_{n+1}}{a_n}$.