$S_l, S_2 \ldots . S_n$ are the sums of infinite geometric series whose first terms are $1,2,3, \ldo

Question

$S_l, S_2 \ldots . S_n$ are the sums of infinite geometric series whose first terms are $1,2,3, \ldots . \mathrm{n}$ and common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4} \ldots \frac{1}{n+1}$ respectively. Then $S_1+S_2+\ldots \ldots+S_n$ equals

Professor PC Thomas Classes · Answer

Using formula 5(a), $S_r=\frac{r}{1-\frac{1}{r+1}}=r+1$ Evaluate $\mathrm{S}_1+\mathrm{S}_2+\ldots . \mathrm{S}_n$ by putting $\mathrm{r}=1,2, \ldots, \mathrm{n}$