0
$\begingroup$

Suppose that $a_n$ is a complex sequence such that $\limsup_{n \to \infty} |a_n|^{1/n} = L$ for some $L$. Then, I want to verify the claim:

$$ \limsup_{n \to \infty} n^{3/n}|a_n|^{1/n} = L$$

But, I am not so sure how to manipulate this sequence. How do I estimate the term $n^{1/n}?$

  • 0
    Use the limit $n^{1/n}\to 1$. – 2017-02-23
  • 0
    @Fnacool Why is that limit equal to 1? – 2017-02-23
  • 0
    Clearly bounded below by $1$. And for upper bound observe that for any $a$ positive, $n<(1+a)^n$ provided $n$ large enough , and take $n$-th root... – 2017-02-23

0 Answers 0