I'm not sure if this is what you want, but there is a theorem that a bounded increasing sequence of selfadjoint operators has a least upper bound, which is the strong limit of the sequence. This result is given in Appendix II of Dixmier's book

*Les algèbres d'opérateurs dans l'espace Hilbertien*. (Dixmier's book is available in an English translation under the title of

*Von Neumann algebras*. The English translation has the added bonus of an extended preface written by me.

)

The key thing here is that the sequence should be increasing (or decreasing and bounded below). For the series $\displaystyle \textstyle\sum_i T_i$ that is of course equivalent to each operator $\displaystyle T_i$ being positive. I'm not sure if that is what you mean by saying that your sum is monotone.

Strong convergence means exactly that $\displaystyle \textstyle\sum_i T_ix\to Tx$ in norm.