In mathematics, an infinite **periodic continued fraction** is a continued fraction that can be placed in the form

where the initial block of *k* + 1 partial denominators is followed by a block of partial denominators that repeats over and over again, *ad infinitum*. For example can be expanded to a periodic continued fraction, namely as .

The partial denominators {*a*_{i}} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of regular continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators *a*_{i} (*i* ≥ 1) are positive integers.

Read more about Periodic Continued Fraction: Purely Periodic and Periodic Fractions, Relation To Quadratic Irrationals, Reduced Surds, Length of The Repeating Block, See Also

### Other articles related to "continued, periodic continued fraction, continued fraction":

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**Periodic Continued Fraction**s

... An infinite

**periodic continued fraction**is a

**continued fraction**of the form where k ≥ 1, the sequence of partial numerators {a1, a2, a3.. ... Ak, and Bk are the numerators and denominators of the k-1st and kth convergents of the infinite

**periodic continued fraction**x, it can be shown that x converges to one of the ... If r1 and r2 are finite then the infinite

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**Periodic Continued Fraction**- See Also

... Hermite's problem

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... During the 19th century, as New York City

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“The mother as a social servant instead of a home servant will not lack in true mother duty.... From her work, loved and honored though it is, she will return to her home life, the child life, with an eager, ceaseless pleasure, cleansed of all the fret and *fraction* and weariness that so mar it now.”

—Charlotte Perkins Gilman (1860–1935)

“But parents can be understanding and accept the more difficult stages as necessary times of growth for the child. Parents can appreciate the fact that these phases are not easy for the child to live through either; rapid growth times are hard on a child. Perhaps it’s a small comfort to know that the harder-to-live-with stages do alternate with the calmer times,so parents can count on getting *periodic* breaks.”

—Saf Lerman (20th century)

“The protection of a ten-year-old girl from her father’s advances is a necessary condition of social order, but the protection of the father from temptation is a necessary condition of his *continued* social adjustment. The protections that are built up in the child against desire for the parent become the essential counterpart to the attitudes in the parent that protect the child.”

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