​if $$\sum a_n$$ be an infinite series of non-negative terms and $$\overline{lim}$$ $${a_n^{1/n}} = p$$, then $$\sum a_n$$ is convegent if​

Correct option is A

​$$\textbf{Given:}\\ \quad \sum a_n \text{ is a series with non-negative terms, and } \lim a_n^{1/n} = p \\[6pt]\textbf{Using Cauchy's Root Test:} \\\text{Let } L = \limsup a_n^{1/n} \\[4pt]\begin{cases}L < 1 & \Rightarrow \text{Series converges} \\L > 1 & \Rightarrow \text{Series diverges} \\L = 1 & \Rightarrow \text{Test is inconclusive}\end{cases} \\[10pt]\text{Since } \lim a_n^{1/n} = p \text{ exists, we apply directly:} \\p < 1 \Rightarrow \sum a_n \text{ converges} \\[10pt]\boxed{\text{Answer: (A) } p < 1}$$​​