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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.
Explanation
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