Partial sum

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.

5 Must Know Facts For Your Next Test

1. 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.
2. Partial sums are used to determine whether an infinite series converges or diverges.
3. If the limit of the partial sums $\lim_{n \to \infty} S_n$ exists and is finite, then the series converges.
4. 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.
5. The concept of partial sums applies to both arithmetic and geometric series.

Review Questions

• What is a partial sum and how is it used in analyzing infinite series?
• How do you denote the $n$-th partial sum of a sequence?
• What condition must be met for an infinite series to converge based on its partial sums?

Related terms

Infinite Series:

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

Convergence:

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

Geometric Series:

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