Partialsumme

Partialsumme, in mathematics, refers to the sum of the first n terms of a sequence. For a sequence (a_k), the nth partial sum is defined by S_n = a_1 + a_2 + ... + a_n. The sequence of partial sums is fundamental in the study of infinite series: the series sum_{k=1}^∞ a_k converges to a finite limit L precisely when the sequence {S_n} converges to L as n tends to infinity.

Examples

- Arithmetic sequence a_k = a_1 + (k-1)d yields S_n = n/2 [2a_1 + (n-1)d].

- Geometric sequence a_k = a_1 r^{k-1} gives S_n = a_1(1 - r^n)/(1 - r) for r ≠ 1; if |r| < 1, S_n converges to a_1/(1 - r) as n → ∞.

Properties

- If all a_k ≥ 0, then the partial sums S_n form a monotonically increasing sequence.

- The behavior of S_n as n grows determines the convergence or divergence of the corresponding series.

- Partial sums are often used to simplify or telescope sums, and to analyze convergence rates and summation methods.

Applications

Partial sums appear across analysis, numerical summation, and probability. They provide a constructive way to define infinite sums, to approximate series numerically, and to study convergence criteria and related theorems such as the comparison test, ratio test, and Abel’s and Dirichlet’s tests. The concept also underpins many algorithms that accumulate quantities step by step.