Re: Limit of Infinitely Nested Expression (4.0 succeeds, 5.2 fails...)

*To*: mathgroup at smc.vnet.net*Subject*: [mg71680] Re: Limit of Infinitely Nested Expression (4.0 succeeds, 5.2 fails...)*From*: bobbaillie at frii.com*Date*: Sun, 26 Nov 2006 03:49:14 -0500 (EST)*References*: <ek97sj$irp$1@smc.vnet.net>

I contacted Wolfram tech support about this bug. They replied that 4.0 was in error, and that 5.2 is working as it should. (!) I don't buy their argument. I think they should fix it, to make it possible to compute the exact limit in this case. Nevertheless, here is their reply: I believe that Mathematica is behaving correctly in this example. Nest and NestList allow you to apply functions a fixed number of times. In this sense, the behavior of Ver. 5.X is correct while the behavior of the package "Limit" for Ver. 4 is inappropriate. Often you may want to apply functions until the result no longer changes. You can do this using FixedPoint. In[2]:= FixedPoint[N[Sqrt[5 + #1] &, 40], N[5, 40]] Out[2]= 2.791287847477920003294023596864004244492 Note that, the following won't terminate. In[3]:= FixedPoint[Sqrt[5 + #1] &, 5] Out[3]= $Aborted because "SameTest" is done symbolically. You may do In[4]:= FixedPoint[Sqrt[5 + #1] &, 5.0000000000000000] Out[4]= 2.791287 though it will take a lot more time if you add more "0"'s to the right of 5.0000000000000000 This is again because of symbolic "SameTest." If you do want to symbolically compute the fixed point of the function, then, as you pointed out, you have to manually solve the equation: In[5]:= Solve[x == Sqrt[5 + x]] Out[5]= 1 + Sqrt[21] {{x -> ------------}} 2