Answer by achille hui for Evaluate the sum to n terms :...
Notice$$\begin{align}\frac{rx^{r-1}}{\prod_{k=1}^r(x+k)}&= \frac{rx^r}{\prod_{k=0}^r(x+k)}= \frac{((x+r)-x)x^r}{\prod_{k=0}^r(x+k)}\\&= \frac{x^r}{\prod_{k=0}^{r-1}(x+k)} -...
View ArticleAnswer by Simply Beautiful Art for Evaluate the sum to n terms :...
Consider the following similar problem$$f_N(x,n)=\sum_{r=1}^N\frac{rx^{r-1}}{(n+1)(n+2)(n+3)\dots(n+r)}=\sum_{r=1}^N\frac{rx^{r-1}n!}{(n+r)!}$$Integrate $f_N(x,n)$ with respect to $x$:$$\int...
View ArticleEvaluate the sum to n terms :...
The main question is :Evaluate the sum to n terms : $$\frac{1}{x+1}+\frac{2x}{(x+1)(x+2)}+\frac{3x^2}{(x+1)(x+2)(x+3)}+\dots$$My approach :My first intuition was to make the series telescopic....
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