(related to Proposition: Infinite Geometric Series)
In order to calculate the sum of partial sums of the geometric series \(\sum_{k=0}^\infty x^n\), we can use the formula of geometric progression and get
\[\sum_{k=0}^n x^n=\frac{1-x^{n+1}}{1-x}.\]
Because, by hypothesis, \(x\) is a real number with an absolute value \(|x| < 1\), it follows from the convergence behavior of the sequence \((x_n)_{n\in\mathbb N}\) that \(\lim_{n\to\infty} x^n=0\). Thus, the limit of the real infinite series is
\[\sum_{k=0}^\infty x^n=\frac{1}{1-x}.\]