       Re: Limits of Nested Expressions

• To: mathgroup at smc.vnet.net
• Subject: [mg80762] Re: [mg80718] Limits of Nested Expressions
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sat, 1 Sep 2007 00:24:02 -0400 (EDT)

```Generalizing to Sqrt[c + Sqrt[c + Sqrt[c + ...]]]

f[c_] = x /. Solve[Sqrt[c + x] == x, x][]

(1/2)*(Sqrt[4*c + 1] + 1)

Looking at values of c that produce an integer

Reduce[f[c] == n, c, Integers]

Element[n | c, Integers] &&
n >= 1 && c == n^2 - n

Simplify[f[n (n - 1)], n >= 1]

n

Table[FixedPoint[Sqrt[n (n - 1) + #] &, RandomReal[]], {n, 10}]

{1.,2.,3.,4.,5.,6.,7.,8.,9.,10.}

Bob Hanlon

---- Bob Hanlon <hanlonr at cox.net> wrote:
> FixedPoint[N[Sqrt[5 + #], 50] &, 5]
>
> 2.7912878474779200032940235968640042444922282883840
>
> x /. Solve[Sqrt[5 + x] == x, x][]
>
> (1/2)*(1 + Sqrt)
>
> N[%, 50]
>
> 2.7912878474779200032940235968640042444922282883840
>
> % == %%%
>
> True
>
>
> Bob Hanlon
>
> ---- Yaroslav Bulatov <yaroslavvb at gmail.com> wrote:
> > Is it possible to compute the following limit in Mathematica 6?
> > Limit[Nest[Sqrt[5 + #]&, 5, n], n -> Infinity]
> >
> > It used to be possible through Calculus`Limit package, which seems to
> > be gone
> >
> >

```

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