Please solve the question in the attachment .... .. ... ANS is e/e-1.......

Actually the above question is superbly hard. I tried my best to solve this but failed miserably. When I searched the internet though, I came across a solution on mathstackexchange. But even that solution relied on approximation, and was not a mathematically rigorous solution. Basically what it does is that it writes the rth term as (1-(n-r)/n)^n and then under the limit approximates it to e^(-n+r), and then taking the sum using gp formula arrives at the reqd limit. You can search it on Google as well, but the problem is that approximating the limit in the above manner is not justified in any manner.