Aid for understanding some calculations

[ShayanJ]

Its a while I'm trying to understand [ http://arxiv.org/pdf/quant-ph/0108132v1.pdf ] but i have problem following its calculations.The part I have problem in now is the expansion in page 14,Eq. 50.Can anyone help me understand how that is done?
Thanks

 

[strangerep]

Its a while I'm trying to understand [ http://arxiv.org/pdf/quant-ph/0108132v1.pdf ] but i have problem following its calculations.The part I have problem in now is the expansion in page 14,Eq. 50.Can anyone help me understand how that is done?
Thanks

Expand $\hat U$ as a Taylor series.

(If you need any more help than that, then please show your work, as if this were a question in the homework forum.)

 

[ShayanJ]

Expand $\hat U$ as a Taylor series.

(If you need any more help than that, then please show your work, as if this were a question in the homework forum.)

I know that.The point is,I don't know how to do such calculations in this abstract manner and without using matrix representations and with that method,I don't seem to get the same result as that of the mentioned paper.
So,this is not actually like a homework thread,because i don't want the exact calculations.I just want an explanation,with some little examples,to understand how should I do it because I can't find a reference which helps me to understand that.
Still,if it seems a homework thread to you,I should say I don't know what to do from the first place,so I can show no work!