# How to get Mathematica to show limit of a recurrence sequence?

I want to find the limit of a such a sequence:

$f(n+1)=\sqrt{2f(n)-1}, \, f(0)=a, a>1$

How does one do this? Is it possible to have it show the solution algebraically, like it does with standard (non-recurrence) limits in WolframAlpha? (I wasn't able to do it on WolframAlpha).

For the recursive formula

$$a_{n+1}=\sqrt{2a_{n}-1}$$

Solving $$\lim_{n\rightarrow \infty} a_n$$

It is equivalent to solve the equation $$x=\sqrt{2x-1}$$

$$\Rightarrow x=1$$

### Update

For fixed-point theory $$x=\phi(x) \quad (x>1)$$

• $$0<|\phi'(x)|=\frac{1}{\sqrt{2x-1}}<\frac{1}{\sqrt{2\times 1-1}}=1$$

• $$x \in [a+1,a+2] \Rightarrow \phi(x)\in [\sqrt{2a+1},\sqrt{2a+3}]\subseteq [a+1,a+2]$$ where $$a>1$$

So for arbitary initial value $$x_0 \in [a+1,a+2]$$, the recursive formula $$x_{k+1}=\phi(x_k) \quad (k=0,1,2 \cdots)$$ is convergent.

Just for fun you can visualize the rate of convergence for various starting position (and the monotony of approach to 1):

f[n_, p_] := Nest[Sqrt[2 # - 1] &, p, n]
Manipulate[
Column[{DiscretePlot[f[j, p], {j, 1, n}, PlotRange -> {0, 10},
GridLines -> {None, {1}}, GridLinesStyle -> Red, ImageSize -> 500],
Join @@ ({#, {Last@#, Last@#}} & /@
Partition[f[#, p] & /@ Range[0, n], 2, 1])},
Plot[{Sqrt[2 x - 1], x}, {x, 0, 20},
Epilog -> {Arrow[ladder], {Red, PointSize[0.02], Point[{1, 1}]}},
ImageSize -> 500]]
}], {n, Range[10, 100, 45]}, {p, 1.1, 20, Appearance -> "Labeled"
}] NestList and ListPlot could have been used instead of DiscretePlot

As an update, I just wanted to add that in Version 11.2 and later, the limit of this sequence can be obtained using RSolveValue by giving a numerical initial value as follows.

In:= RSolveValue[{f[n + 1] == Sqrt[2*f[n] - 1], f == 3}, f[Infinity], n]

Out= 1


Clear[f];


As stated in previous answer, in the limit the equation becomes

eqn = f[n] == Sqrt[2 f[n] - 1];

Solve[eqn, f[n]][]


{f[n] -> 1}

So the limit of the sequence is 1. This can also be demonstrated with FixedPoint; however, since the sequence converges very slowly it is best to start with an initial value (a) very close to 1.

FixedPoint[Sqrt[2 # - 1] &, 1.0001]


1.

EDIT: To demonstrate how slowly this converges

fpl = FixedPointList[Sqrt[2 # - 1] &, 1.0001];

Length[fpl]


76684440

 fpl[[-1]]


1.