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Continued fraction for the natural logarithm

A continued fraction for the natural logarithm is

Here log stands for the natural logarithm.



The power series for the logarithm is

and by substitution we find

From the logarithm of a quotient it follows that

This power series converges for |z| < 1 and therefore you can write this as the sum of products

With Euler's formula for a continued fraction you will find

You can convert this to


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