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

that is of course equivalent to each operator

being positive. I'm not sure if that is what you mean by saying that your sum is monotone.

Strong convergence means exactly that

in norm.