Infinite series and partial sum

A series $\sum a_n$ is the limit of its partial sums $S_N=\sum_{n=1}^N a_n$ as $N\to\infty$ .

An infinite series $\sum_{n=1}^{\infty}a_n$ is defined as the limit of its partial sums $S_N=\sum_{n=1}^{N}a_n$ (the sum of just the first $N$ terms) as $N\to\infty$ . The series CONVERGES to $S$ if $\lim_{N\to\infty}S_N=S$ exists, and DIVERGES otherwise — a series is fundamentally a limit of finite sums, never literally "adding infinitely many numbers."

Examples

For $\sum_{n=1}^{\infty}\frac{1}{2^n}$ , the partial sums $S_1=\frac12,\ S_2=\frac34,\ S_3=\frac78,\ldots$ approach $1$ , so the series converges to $1$ .