A series adds the terms of a sequence. For a sequence $(a_n)$, the finite sum

$$S_N=\sum_{n=1}^{N}a_n$$

is the $N$th partial sum.

An infinite series is not defined by performing infinitely many additions at once. Instead, it is defined through the limit of its partial sums:

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

when that limit exists.

Terms and partial sums are different sequences

The terms $a_n$ and the partial sums $S_N$ play different roles. For example, in

$$1+\frac12+\frac14+\frac18+\cdots,$$

the terms approach zero, while the partial sums

$$1,\quad\frac32,\quad\frac74,\quad\frac{15}{8},\ldots$$

approach $2$. Therefore

$$\sum_{n=0}^{\infty}\left(\frac12\right)^n=2.$$

Geometric series

A geometric series has the form

$$a+ar+ar^2+ar^3+\cdots.$$

When $|r|<1$, its partial sums approach

$$\frac{a}{1-r},$$

so

$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}.$$

When $|r|\ge1$ and $a\ne0$, the geometric series does not converge.

A necessary condition

If a series $\sum a_n$ converges, then its terms must satisfy

$$a_n\to0.$$

The converse is not true: terms approaching zero are necessary but not sufficient for convergence.