Hi jackie,
I think we can get what you're looking for using the Cauchy-Schwarz inequality. Give it a shot with this and let me know how it goes. Good luck!
Thank you for your help GJA. By using the Cauchy-Schwarz inequality I am able to get a bound for the sequence of partial sums of the series . Clearly, the sequence of partial sums is increasing. That should be enough to show convergence right?