infinite series
Imagine trying to add up infinitely many numbers, one after another, and asking whether the running total ever settles down to a single value. That running total is the heart of an infinite series. You do not literally add infinitely many things in one stroke; instead you watch the totals of the first one term, then two, then three, and so on, and ask where they are heading.
Formally, given a sequence a_1, a_2, a_3, ... of real (or complex) numbers, the infinite series is written sum a_n. Its meaning is defined entirely through the sequence of partial sums s_N = a_1 + a_2 + ... + a_N. The series is said to converge to a value S if the sequence s_N converges to S in the usual sense of limits; otherwise the series diverges. So a series is not a number by itself — it is a symbol that may or may not name the limit of its partial sums.
A subtle point worth stating plainly: the symbol sum a_n carries two meanings. It denotes the formal object (the recipe for the partial sums) and, when convergence holds, the resulting limit value. Many manipulations that are automatic for finite sums — reordering terms, grouping them, multiplying two series — can fail or change the answer for infinite series unless extra hypotheses hold.
For a_n = 1/2^n starting at n = 1, the partial sums are 1/2, 3/4, 7/8, 15/16, ... = 1 - 1/2^N, which tend to 1. So sum 1/2^n = 1.
A convergent series whose sum is read off from a closed form for the partial sums.
Convergence of a series is by definition convergence of the sequence of partial sums; every theorem about sequence limits transfers directly to series through this bridge.