Sum of infinite geometric series

If $S_1,S_2,S_3,\dots ,S_p$ are the sums of infinite geometric series, whose first terms are $1,2,3,\dots ,p$, and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots ,\frac{1}{p+1}$ respectively, prove that $S_1+S_2+S_3+\cdots +S_p=\frac{p}{2}(p+3)$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. V. Geometrical Progression, Examples V.b, No. 22