$\displaystyle a_1=1$, $\displaystyle a_{n+1}=a_n+\frac 1{S_n}$ where $\displaystyle n=1,\,2,\ldots$ and $\displaystyle S_n=a_1+a_2+\ldots +a_n$. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_n=\infty$. Take $\displaystyle b_n=n(a_n^2-a_{n-1}^2)$; does it have a limit if yes what?