What Happens to the Limit of an Equation as k Approaches Infinity?

[magma_saber]

Homework Statement

$$\sum(\frac{1}{\sqrt{ln k +2}-\sqrt{ln k -2})}^k$$
as k $$\rightarrow$$$$\infty$$

Homework Equations

Root test: (a_k)^1/k

The Attempt at a Solution

(a_k)^1/k = (\frac{1}{\sqrt{ln k +2}-\sqrt{ln k -2})
does it equal 0? since 1/$$\infty$$ = 0
but its $$\infty$$ - $$\infty$$ i would have to use l'Hôpital's rule right?

 

[quantumdude]

No, I would rationalize the denominator. L'Hopital won't be necessary.

 

[magma_saber]

Then i would get

$$\frac{\sqrt{ln k + 2} + \sqrt{ln k - 2}}{4}$$

as k approaches infinity, the function would also approach infinity so it diverges?

 

[quantumdude]

That's what I got.

 

[magma_saber]

thanks