[tex]U_{n+1} - U_{n} [/tex] = ((1/n+1) + (1/n+2) + ......+ (1/2n+2)) - ((1/n) + (1/n+1) + ...... (1/2n)) = (1/2n+1) + (1/2n+2) - (1/n) = [tex] \frac{2n+2+2n+1}{(2n+1)(2n+2)} [/tex]-[tex] \frac{1}{n} [/tex]= [tex] \frac{4n+3}{(2n+1)(2n+2)} [/tex]-[tex] \frac{1}{n} [/tex] = [tex] \frac{-3n - 2}{n(2n+1)(2n+2)} [/tex]
