Evaluate Limit: $$\lim_{x\to\infty} (-1)^nn^3 + 2^{-n}$$

[tmt1]

I have this limit:

$$\lim_{{x}\to{\infty}} {(-1)}^{n}{n}^{3} + {2}^{-n}$$

and I'm unsure how to evaluate it or how to apply L'hopital's rule to this limit.

 

[soroban]

I have this limit: $$\lim_{n\to\infty} (-1)^n\,n^3 + 2^{-n}$$

and I'm unsure how to evaluate it or how to apply L'hopital's rule to this limit.

L'Hopital's rule doesn't apply here.

We see that: $$\lim_{n\to\infty}2^{-n}\:=\:0$$

But the first part is an alternating sequence of cubes:
. . $$-1 + 8 - 27 + 64 - 125 + \cdots$$ which diverges.

 

[Greg]

It's a limit, not a series - the limit does not exist.