Examples of Convergent and Divergent Series

All infinite series must be either convergent or divergent.

$\sum_{n=1}^\infty a_n = \lim_{k \to \infty} \sum_{n=1}^k a_n$

If $r < 1$, then $\sum_{n=0}^\infty r^n$ is always convergent.

If $\{b_n\}$ is a strictly-decreasing infinite sequence of positive numbers, then $\sum_{n=1}^\infty b_n$ is always convergent.

If $\sum_{n=1}^\infty a_n$ converges to a value $A$, then $\sum_{n=1}^\infty (x \cdot a_n)$ converges to $x \cdot A$ for all real numbers $x$.

If $\sum_{n=1}^\infty a_n$ is divergent, then $\sum_{n=1}^\infty (x \cdot a_n)$ is divergent for all real numbers $x$.