részösszegsorozat

Részösszegsorozat is a Hungarian term that translates to "partial sum sequence" in English. It is a fundamental concept in mathematics, particularly in the study of sequences and series. A partial sum sequence is derived from an initial sequence of numbers. For a given sequence denoted as $a_1, a_2, a_3, \dots$, the partial sum sequence, usually denoted by $S_n$, is defined as the sequence of sums of the first $n$ terms of the original sequence.

The first term of the partial sum sequence, $S_1$, is simply the first term of the original sequence, $a_1$. The second term, $S_2$, is the sum of the first two terms of the original sequence, i.e., $S_2 = a_1 + a_2$. In general, the $n$-th term of the partial sum sequence, $S_n$, is the sum of the first $n$ terms of the original sequence: $S_n = a_1 + a_2 + \dots + a_n$. This can be expressed more compactly using summation notation as $S_n = \sum_{k=1}^{n} a_k$.

The concept of the partial sum sequence is crucial for understanding the convergence of infinite series. An infinite series $\sum_{k=1}^{\infty} a_k$ is said to converge to a limit $L$ if the sequence of its partial sums, $S_n$, converges to $L$ as $n$ approaches infinity. If the partial sum sequence does not converge, the infinite series is said to diverge. Therefore, studying the properties of the partial sum sequence allows mathematicians to determine whether an infinite series will have a finite sum. Examples include geometric series and arithmetic series, where the partial sums can be calculated using specific formulas.