Summations #1

Algebra Level 3

Evaluate \[\begin{align*} 1-\left[\dfrac{1+x}{1+nx}\right]\dbinom{n}{1}+\left[\dfrac{1+2x}{\left(1+nx\right)^2}\right]\dbinom{n}{2}-\left[\dfrac{1+3x}{\left(1+nx\right)^3}\right]\dbinom{n}{3}\\~~~~~~~~~~~~~~~~~~~~~~~~~~+\left[\dfrac{1+4x}{\left(1+nx\right)^4}\right]\dbinom{n}{4}+\cdots+(-1)^n \left[\dfrac{1+nx}{\left(1+nx\right)^n}\right]\dbinom{n}{n}\end{align*}\]

Take \(n = 5,~x = 7\)

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