Kokonaissarja

Kokonaissarja is a term found in Finnish mathematical language that refers to a series, i.e., the sum of the terms of a sequence that are indexed in a prescribed way, typically from n = 1 to infinity. In many contexts it is used to discuss infinite series and their properties, though terminology can vary and the word is not as common as the more general "sarja" or the English "infinite series."

Formally, a kokonaissarja is written as the sum ∑_{n=1}^{∞} a_n, and its behavior is analyzed through the sequence of partial sums S_N = ∑_{n=1}^{N} a_n. If the limit of S_N exists as N approaches infinity, the kokonaissarja is convergent and its value equals this limit. If the limit does not exist or grows without bound, the series diverges. Convergence can be tested using standard criteria such as the ratio test, the root test, the comparison test, or the integral test applied to appropriate examples.

Examples commonly discussed include the geometric series, where a_n = ar^{n-1}; this series converges if |r| < 1 and sums to a/(1 − r). The harmonic series, with a_n = 1/n, is a classic divergent example. Telecoping and power series are also topics that may appear under the broader umbrella of kokonaissarjat in educational materials.

In summary, kokonaissarja denotes a mathematical series and is used to frame discussions of convergence, sums of sequences, and related tests within Finnish mathematics literature. See also sarja (matematiikka) and infinite series.