Does This Sequence Converge or Diverge?

[ironman]

Homework Statement

Show if this sequence (with n=1 to infinity) diverge or converge

Homework Equations

[/B]

The Attempt at a Solution

If I use the Limit Comparison Test:

compare with

so you get

that equals

lim n -> inf

=> inf.

Can I use the Test like this? What does this tell me? (If anything at all...)

 

[Orodruin]

I suggest doing limit comparison with ##(2/3)^n## instead.

 

[ironman]

I suggest doing limit comparison with ##(2/3)^n## instead.

Ah the limit is now 1 .
So:
Limit < infinity.
And ##(2/3)^n## converges.
So the original function must converge as well.

 

[Ray Vickson]

Ah the limit is now 1 .
So:
Limit < infinity.
And ##(2/3)^n## converges.
So the original function must converge as well.

Right, and a direct comparison is easier. First, for ##a > 1## the function ##a^n## always swamps ##n^k## for any fixed power ##k##, meaning that for large enough ##n## we always have ##a^n > n^k##. So, if
[tex] t_n = \frac{2^n + n^2}{3^n + n^3}[/tex]
we have that the numerator is ##2^n + n^2 < 2^n + 2^n = 2 \cdot 2^n## for all ##n > n_0## (where you can even find ##n_0## if you want). The denominator is ##3^n + n^3 > 3^n##, so ##0 < t_n < \frac{2 \cdot 2^n}{3^n} = 2(2/3)^n##, and now you are almost done.