Definition:Properly Divergent/Series

Definition

Let $s$ be an infinite series:

$s = \ds \sum_{k \mathop = 1}^\infty a_k = a_1 + a_2 + a_3 + \cdots$

Then $s$ is a properly divergent series if and only if it is a divergent series such that either:

$s_n \to \infty$ as $n \to \infty$

or:

$s_n \to -\infty$ as $n \to \infty$

where $s_n$ is the $n$th partial sum of $s$:

$s_n = \ds \sum_{k \mathop = 1}^n a_k$

Examples

Natural Numbers

The infinite series:

\(\ds s\)

\(=\)

\(\ds \sum_{k \mathop = 1}^\infty k\)

\(\ds \)

\(=\)

\(\ds 1 + 2 + 3 + 4 + \cdots\)

is a properly divergent series.

Also see

• Definition:Oscillating Series

• Results about properly divergent series can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divergent series
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): properly divergent
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divergent series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): properly divergent