> restart:
> Sn:=factor(1/(N+p-1)*(sum(1/(N+p)^2,k=1..N+p-1)+(1-1/(N+p))^2));

                                     1
                             Sn := -----
                                   N + p

> C:=factor(sqrt(1/(N+p))/(1-1/(N+p)-sqrt(2*log(log(N+p))/(N+p))/(N+p)))
> ;

                                        1/2
                               (N~ + p~)
             C := ---------------------------------------
                                 1/2 /ln(ln(N~ + p~))\1/2
                  N~ + p~ - 1 - 2    |---------------|
                                     \    N~ + p~    /

> plot(subs(N=10,factor(C*(1-1/(N+p)))),p=1..10);

> factor(subs(p=1,C*(1-1/(N+p))));

                                 N~
             -------------------------------------------
                     1/2 /      1/2 /ln(ln(N~ + 1))\1/2\
             (N~ + 1)    |N~ - 2    |--------------|   |
                         \          \    N~ + 1    /   /

> solve(sqrt(log(log(N+1)))>=0,N);1/(N+1)^(1/2)*N/(N-2^(1/2)*(ln(ln(N+1)
> )/(N+1))^(1/2));

                   RealRange(exp(1) - 1, infinity)


                                 N~
             -------------------------------------------
                     1/2 /      1/2 /ln(ln(N~ + 1))\1/2\
             (N~ + 1)    |N~ - 2    |--------------|   |
                         \          \    N~ + 1    /   /

> plot(sqrt(log(log(N+1))),N=0..10);

> 
