[tex]f(x) = x+x^2+x^3+...+x^n-n \\ f(1) = 1+1^2+1^3+...+1^n-n \\ f(1) = 1+1+1+\underset{\text{de n ori}}{\underbrace{...}}+1 - n \\ f(1) = 1\times n-n \\ f(1) = 0 [/tex]
[tex]\lim\limits_{x\rightarrow 1}\dfrac{x+x^2+x^3+...+x^n-n}{x-1} \quad \overset{\boxed{\dfrac{0}{0}} \rightarrow L'H}{=} \\ \\ = \lim\limits_{x\rightarrow 1}\dfrac{(x+x^2+x^3+...+x^n-n)'}{(x-1)'} = \\ \\ = \lim\limits_{x\rightarrow 1}\dfrac{1+2x+3x^2+....+nx^{n-1}}{1} = \\ \\ = 1+2\cdot 1+3\cdot 1^2+....+n\cdot 1^{n-1} = \\ \\ = 1+2+3+...+n = \\ \\ = \dfrac{n (n+1)}{2}[/tex]
