Definition of Ad infinitum

In context, it usually means "continue forever," and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever", among other uses. It may also be used in a manner similar to the Latin phrase "et cetera" to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:

• "The sequence 2, 4, 6, 8, 10 ... continues ad infinitum."
• "The perimeter of a fractal may be iteratively drawn ad infinitum."
• The 17th century writer Jonathan Swift mocked the idea of self-similarity in natural philosophy with the following ditty:

"So nat'ralists observe, a flea

Hath smaller fleas that on him prey,

And these have smaller fleas that bite 'em,

And so proceed ad infinitum."

• Which was itself parodied by the Victorian era mathematician Augustus De Morgan:

"Great fleas have little fleas upon their backs to bite 'em,

And little fleas have lesser fleas, and so ad infinitum.

And the great fleas themselves, in turn, have greater fleas to go on,

While these again have greater still, and greater still, and so on."