they may converge to a finite limit, or even oscillate among multiple finite limits. No matter how the finite convergents behave, the continued fraction defined by the sequence diverges by oscillation with the point at infinity in this case. The sequence may produce no more than a finite number of zero denominators . while the subsequence of finite convergents dances wildly around the plane in a pattern that never repeats itself and never approaches any finite limit either. Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction where is any real number such that . Euler's continued fraction formula Euler proved the following identity: From this many other results can be derived, such as and Euler's formula connecting continued fractions and series is the motivation for the , and also the basis of elementary approaches to the convergence problem. Examples Transcendental functions and numbers Here are two continued fractions that can be built via Euler's identity. Here are additional generalized continued fractions: This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s. Example: the natural logarithm of 2 (= ≈ 0.693147...): Here are three of 's best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting and multiplying by 4. The Leibniz formula for : converges too slowly, requiring roughly terms to achieve correct decimal places. The series derived by Nilakantha Somayaji: is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both converge sublinearly to . On the other hand: converges linearly to , adding at least three digits of precision per four terms, a pace slightly faster than the arcsine formula for : which adds at least three decimal digits per five terms. Note: this continued fraction's rate of convergence tends to , hence tends to , whose common logarithm is . The same (the silver ratio squared) also is observed in the unfolded general continued fractions of both the natural logarithm of 2 and the th root of 2 (which works for any integer ) if calculated using . For the folded general continued fractions of both expressions, the rate convergence , hence , whose common logarithm is , thus adding at least three digits per two terms. This is because the