I am studying about series and confused when doing this question.
Let $(a_i)_{i=1}^{\infty}$ be a sequence of real numbers such that $a_i\ge0$ and $\lim_{n\rightarrow \infty}\sup\ (a_n)^\frac1n=L\lt1$, show that $\sum_{n=1}^\infty a_n$ converges.
The question has a hint which is using comparison test.
I don't understand why L doesn't equal 1 since for any b$\gt$0, $\lim_{n\rightarrow \infty}\ (b)^\frac1n=1$ and I have zero idea about how to form a new series to use comparsion test.