Determining whether a sum converges or diverges .

[Timebomb3750]

Determining whether a sum converges or diverges...

Homework Statement

Ʃ((3n!)/(4^(n)))

Homework Equations

I figured I'd do the ratio test with this one.

The Attempt at a Solution

So that would be (3(n+1)!)/(4^(n+1)) * (4^(n))/(3n!), I then cross cancel until I'm left with this...

((n+1)!)/(4n!) Then you can break up the (n+1)!, to cancel a (n!) in the numerator and denominator, so you're left with the limit as n→∞ of (n+1)/(4), but this doesn't look right to me. Where did I screw up?

 

[SammyS]

Homework Statement

Ʃ((3n!)/(4^(n)))

Homework Equations

I figured I'd do the ratio test with this one.

The Attempt at a Solution

So that would be (3(n+1)!)/(4^(n+1)) * (4^(n))/(3n!), I then cross cancel until I'm left with this...

((n+1)!)/(4n!) Then you can break up the (n+1)!, to cancel a (n!) in the numerator and denominator, so you're left with the limit as n→∞ of (n+1)/(4), but this doesn't look right to me. Where did I screw up?

(n+1)/(4) looks right to me !

 

[Timebomb3750]

(n+1)/(4) looks right to me !

But I'm confused regarding the limit of this. Is the limit as n→∞=(1/4)? Well, it becomes (∞/4), so does that diverge. I always get confused when you have infinity on the top, but a constant on the bottom.

 

[SammyS]

But I'm confused regarding the limit of this. Is the limit as n→∞=(1/4)? Well, it becomes (∞/4), so does that diverge. I always get confused when you have infinity on the top, but a constant on the bottom.

The ratio goes to ∞ .