sequences and series - Find $limlimits_{nrightarrowinfty} sumlimits_{k=1}^{n} frac{a_k}{a_k+a_1+a_2+...+a_n}$

Let $(a_n)_{n\geq1}$ a sequence strictly increasing of real positive numbers such that $\lim\limits_{n\rightarrow\infty} \frac{a_{n+1}}{a_n}=1$, find $\lim\limits_{n\rightarrow\infty} \sum_{k=1}^{n} \frac{a_k}{a_k+a_1+a_2+...+a_n}$. I know this should be solved using Riemann integration, but my only significant progress wwas the finding of the partition $0\leq\frac{a_1}{a_1+...+a_n}\leq\frac{a_1+a_2}{a_1+...+a_n}\leq...\leq\frac{a_1+...+a_n}{a_1+...+a_n}=1$ for the interval$[0,1]$.