This gives us that \[ \expec(T_n) = \sum_{k=1}^n \expec(X_{n,k}) = \sum_{k=1}^n \frac{n}{n-k+1} = n\left(\frac{1}{n}+\frac{1}{n-1}+\ldots+\frac{1}{2}+\frac{1}{1}\right) = n H_n, \] where \( H_n = \sum...This gives us that \[ \expec(T_n) = \sum_{k=1}^n \expec(X_{n,k}) = \sum_{k=1}^n \frac{n}{n-k+1} = n\left(\frac{1}{n}+\frac{1}{n-1}+\ldots+\frac{1}{2}+\frac{1}{1}\right) = n H_n, \] where \( H_n = \sum_{k=1}^n 1/k\) is the \emph{$n\)-th harmonic number}, and \[ \var(T_n) = \sum_{k=1}^n \var(X_{n,k}) = \sum_{k=1}^n \frac{k-1}{n}\left(\frac{n}{n-k+1}\right)^2 \le n^2 \sum_{k=1}^n \frac{1}{k^2} \le 2 n^2 $$ (in this example, we only need a bound for \(\var(T_n)\), but it is possible also to get mor…
