Osittaissumma

Osittaissumma is the Finnish term for a partial sum, referring to the sum of the first n terms of a sequence or series. Given a sequence a1, a2, ..., the nth partial sum is S_n = a1 + a2 + ... + a_n. For a series ∑_{k=1}^∞ a_k, the sequence of partial sums S_n is studied; the series is said to converge if lim_{n→∞} S_n exists and is finite, and this limit, when it exists, is the sum of the series.

Properties and uses: Partial sums are used to analyze convergence and to approximate the value of infinite sums. If every term a_k is nonnegative, the sequence of partial sums S_n is nondecreasing. Partial sums also arise in estimations and in proofs, such as proving convergence or establishing bounds.

Examples: For a geometric series with a_k = ar^{k-1} (r ≠ 1), the partial sum is S_n = a(1 − r^n)/(1 − r). If |r| < 1, the series converges and S_n → a/(1 − r) as n → ∞. In a telescoping series where a_k = b_k − b_{k+1}, the partial sum is S_n = b_1 − b_{n+1}, which often allows immediate evaluation of the limit.

Relation to infinite series: The partial sums form a sequence that determines the behavior of the entire series. Convergence of the partial sums implies the series has a finite sum; divergence means the series does not have a finite sum.