Partial sum - (Calculus II) - Vocab, Definition, Explanations | Fiveable

Definition

A partial sum is the sum of the first $n$ terms in a sequence. It provides an approximation to the sum of an infinite series.

The $n$-th partial sum is denoted as $S_n = \sum_{k=1}^{n} a_k$, where $a_k$ are the terms of the sequence.
Partial sums are used to determine whether an infinite series converges or diverges.
If the limit of the partial sums $\lim_{n \to \infty} S_n$ exists and is finite, then the series converges.
For a geometric series with ratio $|r| < 1$, the partial sum can be calculated using $S_n = a \frac{1-r^n}{1-r}$, where $a$ is the first term.
The concept of partial sums applies to both arithmetic and geometric series.

A sum of infinitely many terms, written as $\sum_{k=1}^{\infty} a_k$. Convergence depends on the behavior of its partial sums.

A property of an infinite series where its partial sums approach a finite limit.

A series with a constant ratio between successive terms, expressed as $\sum_{k=0}^{\infty} ar^k$.