Infinite Series and Convergence

An infinite series $\sum_{n=1}^{\infty} a_n$ is defined as the limit of its partial sums $S_N = a_1 + a_2 + \dots + a_N$. The series converges to $S$ exactly when the sequence $(S_N)$ converges to $S$; otherwise it diverges.

The cleanest case is the geometric series $\sum_{n=0}^{\infty} a\,r^{n}$. When $|r| < 1$ it converges, with the closed form

When $|r| \ge 1$ the terms do not shrink to $0$ and the series diverges.

Worked example. $\displaystyle \sum_{n=0}^{\infty} \left(\tfrac12\right)^{n} = 1 + \tfrac12 + \tfrac14 + \dots$ has $a = 1$, $r = \tfrac12$, so the sum is $\dfrac{1}{1 - \frac12} = 2$.

Two workhorse tests: the $n$th-term test (if $a_n \not\to 0$, the series diverges) and the $p$-series fact ($\sum \frac{1}{n^p}$ converges iff $p > 1$). The harmonic series $\sum \frac1n$ is the famous knife-edge: its terms shrink to $0$, yet it diverges.