# Calculate double summation?

GROUPS:
 Hey guys, I need to calculate a double infinite summation. The numerical result seems to be OK when the input are small, while becomes terrible when inputs become larger. The result of EY, which is the expectation of a positive random variable Y, should be strictly positive.1 I am wondering how Mathematica would perform calculation to an infinite summation like this. I tried summation up to 500, 1000, 2000 and it seems that there is no difference in summation after 2000. So what is a reasonable truncation for infinity?2 I tried to increase the precision, up to 1000, while there is no change in the result. 3 The result does change if you move the second summation immediately after the first summation. I do not know the reason for that.4 Is there any quicker way to do the calculation for TA[[1.3]], Tb1[[3,3]] and Tb2[[3,3]].Please see my attachment for more details.Thanks in advance. Attachments: