Részösszeg

Részösszeg is a Hungarian term that translates to "partial sum" in English. It refers to the sum of a subset of terms in a sequence or series. In mathematics, when dealing with a series, a partial sum is the sum of the first n terms. For example, if a series is defined as $a_1 + a_2 + a_3 + \dots$, the first partial sum is $S_1 = a_1$, the second partial sum is $S_2 = a_1 + a_2$, and the nth partial sum is $S_n = a_1 + a_2 + \dots + a_n$.

The concept of partial sums is fundamental to understanding the convergence of infinite series. If the sequence of partial sums $S_n$ converges to a finite limit L as n approaches infinity, then the series is said to converge to L. Otherwise, the series diverges. This allows mathematicians to assign a value to an infinite sum, provided that the partial sums approach a specific number.

In practical applications, partial sums are used in various fields, including numerical analysis, signal processing, and finance. For instance, in financial calculations, partial sums might represent cumulative amounts over specific periods. In numerical methods, approximating a function using a series expansion often involves calculating and summing a finite number of terms, effectively using partial sums. The term itself emphasizes that it is not the total sum but a component of it.