Question

Find the sum of the following series $\frac{n(1+x)}{1+nx}-\frac{n(n-1)}{\lfloor 2}\cdot \frac{1+2x}{(1+nx{)}^2}+\frac{n(n+1)(n-2)}{\lfloor 3}\cdot \frac{1+3x}{(1+nx{)}^3}-....$ to $n$ terms.

Answer 1

$\Rightarrow$ Sum can be expressed in two series;-

$S_1=\frac{n}{1+nx}-\frac{n(n-1)}{2!}\frac{1}{{(1+nx)}^2}+\frac{n(n-1)(n-2)}{3!}\frac{1}{{(1+nx)}^3}$

$S_2=\frac{nx}{1+nx}\{1-\frac{(n-1)}{1+nx}+\frac{(n-1)(n-2)}{2!}\frac{1}{{(1+nx)}^2}\}$

$S_1=1-{(1-\frac{1}{1+nx})}^n$

$S_2=\frac{nx}{1+nx}{(1-\frac{1}{1+nx})}^{n-1}$

$S=S_1+S_2$

$=1-{\left(\frac{nx}{1+nx}\right)}^n+{\left(\frac{nx}{nx+1}\right)}^n$

$S=1$