<s>[INST] Sum of $1/(x+1) +2x/(x+1)(x+2) +3x^2/(x+1)(x+2)(x+3)+\ldots$ till n terms Sum of $$\frac{1}{(x+1)}+\frac{2x}{(x+1)(x+2)}+\frac{3x^2}{(x+1)(x+2)(x+3)}+\ldots$$ till n terms ...
I have no clue how to solve this ... I tried multiplying quantities ... such as $(x+2)-(x+1)$
 [/INST] We have
$$ \frac{kx^{k-1}}{(x+1)(x+2)\cdots(x+k-1)(x+k)}\\= \frac{x^{k-1}}{(x+1)(x+2)\cdots(x+k-1)}- \frac{x^{k}}{(x+1)(x+2)\cdots(x+k)},\, k\geq2$$
so by telescoping
$$\sum_{k=1}^{n} \frac{kx^{k-1}}{(x+1)(x+2)\cdots(x+k-1)(x+k)}=\frac{1}{x+1}+\frac{x}{x+1}-\frac{x^{n}}{(x+1)(x+2)\cdots(x+n-1)(x+n)}\\=1-\frac{x^{n}}{(x+1)(x+2)\cdots(x+n-1)(x+n)}$$
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