Euler's continued fraction formula

English

Named after the 18th-century Swiss mathematician Leonhard Euler (1707-1783).

Euler's continued fraction formula

(mathematics) A formula connecting a finite sum of products to a finite continued fraction: ${\\displaystyle a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\\cdots +a_{0}a_{1}a_{2}\\cdots a_{n}={\\cfrac {a_{0}}{1-{\\cfrac {a_{1}}{1+a_{1}-{\\cfrac {a_{2}}{1+a_{2}-{\\cfrac {\\ddots }{\\ddots {\\cfrac {a_{n-1}}{1+a_{n-1}-{\\cfrac {a_{n}}{1+a_{n}}}}}}}}}}}}}\\,.}$

Euler's continued fration formula also applies in the infinite case:

{\\displaystyle 1+\\sum _{i=1}^{\\infty }\\left(\\prod _{j=1}^{i}r_{j}\\right)={\\cfrac {1}{1-{\\cfrac {r_{1}}{1+r_{1}-{\\cfrac {r_{2}}{1+r_{2}-{\\cfrac {r_{3}}{1+r_{3}-\\ddots }}}}}}}}.}

单词	Euler's continued fraction formula
释义	Euler's continued fraction formula English Etymology Named after the 18th-century Swiss mathematician Leonhard Euler (1707-1783). Proper noun Euler's continued fraction formula (mathematics) A formula connecting a finite sum of products to a finite continued fraction: ${\\displaystyle a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\\cdots +a_{0}a_{1}a_{2}\\cdots a_{n}={\\cfrac {a_{0}}{1-{\\cfrac {a_{1}}{1+a_{1}-{\\cfrac {a_{2}}{1+a_{2}-{\\cfrac {\\ddots }{\\ddots {\\cfrac {a_{n-1}}{1+a_{n-1}-{\\cfrac {a_{n}}{1+a_{n}}}}}}}}}}}}}\\,.}$ Usage notes Euler's continued fration formula also applies in the infinite case: ${\\displaystyle 1+\\sum _{i=1}^{\\infty }\\left(\\prod _{j=1}^{i}r_{j}\\right)={\\cfrac {1}{1-{\\cfrac {r_{1}}{1+r_{1}-{\\cfrac {r_{2}}{1+r_{2}-{\\cfrac {r_{3}}{1+r_{3}-\\ddots }}}}}}}}.}$
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