calculus - convergence of $a_n$ knowing that $a_n^n$ converges

One can prove the following statement:

Let
$(1) a_1, a_2, a_3, ... a_n, ...$
be a sequence of non-negative numbers.
Let also $k>1$ be an integer.

If the sequence
$(2) a_1^k, a_2^k, a_3^k, ... a_n^k, ...$
converges and its limit is $a$, then the sequence (1) also converges and its limit is $\sqrt[k]{a}$.

But what if instead of (2) we know that the sequence
$(3) a_1^1, a_2^2, a_3^3, ... a_n^n, ...$
converges and its limit is b. Can we then state something about (1), and about its convergence, and possibly about its limit (if such a limit exists)?

Answer

Take $(a_n)$ defined by :

$a_n = 0$ if n is even.

$a_n = \frac{1}{2}$ if n is odd.

This sequence satisfies (3) but it does not converge.