Using root test and ratio test for divergence

[superdave]

Homework Statement

Does this series converge or diverge?

Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1)

Homework Equations

The Attempt at a Solution

Okay, I've tried it both ways.

Ratio test:

lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1))

Now, that doesn't appear to simplify in anyway that would make using l'hospital's rule possible to find the limit.

Root test:

lim n --> inf. of -3*n^(1/n) / 4^((n-1)/n)

That bottom part throws me off.

 

[d_leet]

Why does it through you off? What is the limit of (n-1)/n as n goes to infinity?

 

[HallsofIvy]

Homework Statement

Does this series converge or diverge?

Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1)

Homework Equations

The Attempt at a Solution

Okay, I've tried it both ways.

Ratio test:

lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1))

Now, that doesn't appear to simplify in anyway that would make using l'hospital's rule possible to find the limit.

Seriously? You are aware, are you not, that (-3)^(n+1)/(-3)^n= -3? The (4^n part is just as easy! You should not need L'Hospital's rule.

Root test:

lim n --> inf. of -3*n^(1/n) / 4^((n-1)/n)

That bottom part throws me off.