Metod 1:
$$U=\dfrac{x + x^2 + \dots + x^n - n}{x - 1} = \dfrac{\displaystyle\sum_{i=1}^n(x^i-1)}{x-1}$$$$=$$$$\displaystyle\sum_{i=1}^n\dfrac{(x^i-1)}{x-1}=\displaystyle\sum_{i=1}^n(x^{i-1}+x^{i-2}+...+x^2+x+1)$$
$$\lim\limits_{x \to 1}U=\lim\limits_{x \to 1}\displaystyle\sum_{i=1}^n(\underbrace{x^{i-1}+...x^2+x+1}_{i\; terim})=\displaystyle\sum_{i=1}^ni=\dfrac{n(n+1)}{2}$$
Metod 2 için ipucu: türev