When covering issues related to error estimates in a calculus course, students find the technique of making estimates (definition of limit, Newton's method, numerical integration, remainder formula for power series) quite difficult even in basic situations. I suspect they would all accept the following fake theorem of error analysis: if you apply a recursion and two successive terms have the same first n digits then the sequence converges and those common initial digits are the first n digits of the limit.
This is of course bogus, and a slowly divergent process like the partial sums of the harmonic series provides a counterexample. What I would like to have available to show the class are examples of convergent recursions from a typical (not artificially created) example of the kind met in a first-year calculus course. A parametric family of examples would be even better, but interesting individual examples would be good too. It would be especially nice if the example has terms agreeing to the first 6 or so digits but the limit is completely different than what the data suggest.