I'm embarrassed to admit how long it took me to find the formula $\displaystyle x_n = n\sum_{r=1}^n\frac1r$, having previously said that you should be able to guess it after working out the first few terms. For n=1,2,3,4,5 these are $\displaystyle 1,\ 3,\ 5\tfrac12,\ 8\tfrac13,\ 11\tfrac5{12}$. It seemed natural to write these as fractions with factorials as the denominators, namely $\displaystyle \frac1{0!},\ \frac3{1!},\ \frac{11}{2!},\ \frac{50}{3!},\ \frac{274}{4!}$. I still didn't recognise the numerators, so I searched for the sequence 3, 11, 50, 274 in the wonderful

Online Encyclopedia of Integer Sequences (one of the most useful mathematical resources on the internet), and it pointed me in the direction of the harmonic series $\displaystyle \textstyle\sum\frac1r$.