Create a program which solves recurrences, the limiting value of a recurrence relation repeated an infinite number of iterations, either analytically or numerically to the desired precision. (Probably already exists.) Of course, it doesn't work all the time. It can evaluate infinite series which have a closed form.
Inspiration is a magic programmable calculator which can seemingly perform an infinite number of arithmetic operations in finite time. (Though that's what sqrt, exp, and the trigonometric functions already do.)
Would such a subroutine be useful as a compiler optimization? Infinite loops could terminate.
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