If $|a_1|>1$, then the series $\sum\frac{a_1^n+\cdots+a_k^n}{n}$ diverges

Suppose that the series is convergent. Then the power series $$ f(z)=\sum_{k=1}^{\infty} \frac{a_1^k+\cdots + a_n^k}k z^k $$ is valid on $D=\{z: |z|<1\}$.

By differentiation, we have $$ f'(z)=\sum_{k=1}^{\infty} (a_1^k + \cdots + a_n^k)z^{k-1}$$ is valid on $D$.

Moreover the series on the right side equals $$\frac{a_1}{1-a_1z} + \cdots +\frac{a_n}{1-a_n z} $$ on $|z|<\min \{1/|a_i| : 1\leq i\leq n\}<1$ since each term is geometric series.

This function has a pole at $1/a_1$ which is inside $D$. This is a contradiction. Thus, the series must be divergent.