What is and what represents a convergents function in polynomial form?

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$$\mathbf{convergents}(cos(1), 20)$$

What exactly is a convergents function and what, that series of fractions is representing ? There is an use for this in numerical linear algebra ?

Feel free to replace the pair function & value from $cos(1)$ if you need to to explain this .

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Your call produced the first 20 convergents of the (irrational) number $\cos 1$, i.e. rational approximations to it obtained by the theory of continued fractions. Trivia: If you try the golden ration instead of $\cos 1$, you may notice that Fibonacci numbers occur as numerators and denominators.