find all the partial limits of this sequence

Question

i need to find all the partial limits (is that the current terminology? i mean the limits of all possible subsequences for a given sequence)

$a_n = \sqrt[n]{4^2 + 2^n} $

find all partial sums,also find $\liminf_{n\rightarrow\infty} a_n$ and $\limsup_{n\rightarrow\infty} a_n$

i couldn't really find any similar questions with regards to these types of questions, if somone could also point me to what to google for that would be great!

Answer 1

This sequence's limit was asked just yesterday. Here's my answer (hint, in fact):

$$\sqrt[n]{2^n}\le\sqrt[n]{16+2^n}\stackrel{\text{For}\;n\ge 4}\le\sqrt[n]{2\cdot 2^n}$$