Answer
NO
Work Step by Step
The answer is NO.
Consider: $\Sigma_{n=1}^\infty a_n$ is a convergent series of positive numbers.
Then, we have $2\Sigma_{n=1}^\infty a_n= \Sigma_{n=1}^\infty 2a_n$ also converges and $2a_n \geq a_n$
Thus, there will be no largest convergent series of positive numbers.
Hence, the answer is NO.
