In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. Mathematical analysis is largely taken up with studying the conditions under which a given function will result in a convergent infinite series. Such series (e.g., the Fourier series) are particularly useful in solving differential equations.
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An example of an infinite cycle can be found in the children's song "John Jacob Jingleheimer Schmidt". The lyrics and melody don't come to a natural stopping point; those who teach it carry it on by resuming at the start, and the effect is the same as an infinite loop in computing.
One of the most common examples of an infinite recursion is a story. The words vary from telling to telling, but it goes something like this:
In fiction, Eric Rücker Eddison's fantasy novel The Worm Ouroboros implies the same motif simply by having the final situation match the initial one, and having the first paragraph of the physical document precisely duplicated as the final paragraph. It has been published with cover art depicting an Ouroboros, in another evocation of the motif.
James Joyce's novel Finnegans Wake starts in mid-sentence: "riverrun, past Eve and Adam's, from swerve of shore to bend of bay, brings us by a commodius vicus of recirculation back to Howth Castle and Environs". The beginning of this sentence is found in the novel's last words: "A way a lone a last a loved a long the". Thus, this novel is another example of infinite loop.
The works of M. C. Escher contain many variations of both infinite cycles and infinite recursion.
An excellent example is the very popular children's song There's a Hole in My Bucket.
The opening lyrics are as follows (the most common mention Henry and Liza):
Henry: There's a hole in the bucket, dear Liza, dear Liza, There's a hole in the bucket, dear Liza, a hole.
Liza: Well fix it dear Henry, dear Henry, dear Henry, well fix it dear Henry, dear Henry, fix it
Henry: With what shall I fix it, dear Liza, dear Liza, with what shall I fix it dear Liza, with what?
Liza: With straw, dear Henry, dear Henry, dear Henry, with straw, dear Henry, dear Henry, with straw.
This goes on in the same format with different lines:
But the straw is too long.
Then cut it.
With what shall I cut it?
With an axe
But the axe is too dull.
Well sharpen it
How shall I sharpen it
With the stone
But the stone is too dry
Then whet it
With what shall I whet it?
With the water
But I have no water
Then get some
With what shall I get it?
With the bucket
But there's a hole in the bucket!