Struggling with one step to show quantum operator equality

[troglodyte]

Hello guys,

I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work permanently with one Infinite sum and one finite sum.In the solution there is a index shift before the last step.Maybe this is the reason why suddenly one more infinite sum pops out.It would be nice if you can explain to me from where this second infinite sum comes from .
Thank you

 

[DrClaude]

You forgot the picture...

 

[troglodyte]

You forgot the picture...

Thank you for your reply.
I thought I wouldn't get any reply so I have deleted them to reduce server costs.