Explaining Convergence of (3^n)/(2^n + 4^n) w/o Limit Comp. Test

[Erind]

Homework Statement

Use the comparison test to explain why the series (3^n)/(2^n + 4^n) is convergent. He said specifically not to use the limit comparison test on this one.

Homework Equations

1/2^n

The Attempt at a Solution

I know I should be comparing it to 1/2^n because it is a geometric series and thereby convergent, but I can't get rid of the 3^n on the top in order to get there.

 

[Erind]

Ok, I just came up with comparing (3^n)/(2^n + 4^n) < (3^n)/(2^n + 4^n)(3^n) and then canceling the (3^n) and then 1/(2^n + 4^n) < 1/(2^n) and then it's done, but I'm not sure if that first inequality is a legal move.

 

[micromass]

No, I'm afraid that it isn't a legal move

Maybe you should delete some of the terms in the denumerator...

 

[Bohrok]

Try starting with 2ⁿ + 4ⁿ > 4ⁿ, then rewrite the inequality until you get 3ⁿ/(2ⁿ + 4ⁿ) for the left side of the inequality.