partial sum

A partial sum in mathematics is the sum of a finite number of terms from an infinite series. For an infinite series, denoted as (a_1 + a_2 + a_3 + \dots), a partial sum, (s_n), is calculated by adding the first (n) terms: (s_n = a_1 + a_2 + \dots + a_n).1

The concept of partial sums is crucial for understanding whether an infinite series converges or diverges. If the partial sums approach a specific, fixed number as more terms are added, the series converges to that number. If the partial sums do not settle on a single value, the series diverges. For example, the partial sums of the series (\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots) get progressively closer to 1, indicating convergence. In contrast, the partial sums of (1 - 1 + 1 - 1 + \dots) alternate between 0 and 1, showing divergence.12