Infinite Sequences and Series · Hard problem

An accumulated error after $n$ steps is $\displaystyle a_n=\sum_{k=1}^{n}\frac{1}{n^2+k}$ (a sum of $n$ terms, each term using the SAME $n$ in its denominator's base). Using the bound $a_n\le\dfrac{n}{n^2+1}$, find the smallest $n$ for which this bound guarantees $a_n<0.01$.