Natural Logarithm - Continued Fractions

Continued Fractions

While no simple continued fractions are available, several generalized continued fractions are, including:

\ln (1+x)=\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots= \cfrac{x}{1-0x+\cfrac{1^2x}{2-1x+\cfrac{2^2x}{3-2x+\cfrac{3^2x}{4-3x+\cfrac{4^2x}{5-4x+\ddots}}}}}
\ln \left( 1+\frac{2x}{y} \right) = \cfrac{2x} {y+\cfrac{x} {1+\cfrac{x} {3y+\cfrac{2x} {1+\cfrac{2x} {5y+\cfrac{3x} {1+\ddots}}}}}} = \cfrac{2x} {y+x-\cfrac{(1x)^2} {3(y+x)-\cfrac{(2x)^2} {5(y+x)-\cfrac{(3x)^2} {7(y+x)-\ddots}}}}

Read more about this topic:  Natural Logarithm

Famous quotes containing the word continued:

    ““Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)