Finding the limit of this series

Finding the limit of this series

Find the limit of the following series: $$
S_n=\frac1{1+a}+\frac1{2(1+2a)}+\frac1{3(1+3a)}+\cdots$$ where $$
a=\frac12,\frac13,\frac14,\ldots $$
Thanks for your help.