Maybe because there is not any analytical formula which gives this sum. Can this function be integrated in an analytic way? I suppose no.
You can take the numerical result of the sum in this way:
k=1:10000;
sum1=sum(((k+1).^(1/2) - (k-1).^(1/2))./k.^2);
k=1:100000;
sum2=sum(((k+1).^(1/2) - (k-1).^(1/2))./k.^2);
and compare sum1 and sum2. The result is 1.763306701885209.
