Euler's Continued Fraction Formula - The Original Formula

The Original Formula

Euler derived the formula as an identity connecting a finite sum of products with a finite continued fraction.

a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \cfrac{a_0}{1 - \cfrac{a_1}{1 + a_1 - \cfrac{a_2}{1 + a_2 - \cfrac{\ddots}{\ddots \cfrac{a_{n-1}}{1 + a_{n-1} - \cfrac{a_n}{1 + a_n}}}}}}\,

The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite continued fraction.

Read more about this topic:  Euler's Continued Fraction Formula

Famous quotes containing the words original and/or formula:

    “In the end, for congenial sympathy, for poetry, for work, for original feeling and expression, for perfect companionship with one’s friends—give me the country.”
    —D.H. (David Herbert)

    “I cannot give you the formula for success, but I can give you the formula for failure—which is: Try to please everybody.”
    —Herbert B. Swope (1882–1958)