Identity for continued fractions

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With one term you can develop a fraction

but that is still no continued fraction. With two terms that is possible, because

and that can be written as

With three terms you see for the first time occur an iteration

We treat b (1 + c) separately, and write the continued fraction for the time being as

Now we divide the numerator and denominator in the relevant fraction by 1 + c and get

In the denominator of that same fraction we add 1 and subtract it immediately again

so that

For clarity we continue with four terms and notice the iteration

Here we treat c (1 + d) separately, and write the continued fraction as

We now divide by 1 + d and find

This scheme repeats itself over and over again.

This formula was developed by the Swiss mathematician Leonhard Euler (1707 - 1783).