Sum to 10 terms of the series $1+2(1 \cdot 1)+3(1 \cdot 1)^2+4(1 \cdot 1)^3+$ $\_\_\_\_$ is

$S_n=1 \cdot 3+2 \cdot 2^2+3 \cdot 3^3+4 \cdot 2^4+\ldots \ldots \ldots$ upto $n$ terms. If $S_{20}=a \cdot 3^{21}+b \cdot 2^{22}+\frac{391}{288}$, then value of $32 a-9 b$ is

The sum of the series $1 \cdot 2^2+2 \cdot 4^2+3 \cdot 6^2+\ldots$ upto 10 terms is

Sum of first ' $n$ ' terms of a series $a_1+a_2+\ldots+a_n$ is given by $S_n=\frac{n\left(n^2-1\right)(n+2)}{4}$, then the value of $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=2}^n \frac{1}{a_r}$ is