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## Main Question or Discussion Point

The other day I was playing with my calculator and noticed that

[tex]\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} \approx 2[/tex]

But, what is that kind of expression called? How does one justify that limit?

And, to what number exactly does converge, for example:

[tex]\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}} \approx 1.6161[/tex]

[tex]\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+...}}}} \approx 2.3027[/tex]

Any references where I could read about these subjects?

Another question. Considering real [tex]x>1[/tex], we have:

[tex]\Gamma(x) - x^1 = 0[/tex] then [tex] x \approx 2[/tex]

But how does one justify that? And what are the exact values of these functions:

[tex]\Gamma(x) - x^2 = 0[/tex] then [tex] x \approx 3.562382285390898[/tex]

[tex]\Gamma(x) - x^3 = 0[/tex] then [tex] x \approx 5.036722570588711[/tex]

[tex]\Gamma(x) - x^4 = 0[/tex] then [tex] x \approx 6.464468490129385[/tex]

Thanks,

Damián.

[tex]\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} \approx 2[/tex]

But, what is that kind of expression called? How does one justify that limit?

And, to what number exactly does converge, for example:

[tex]\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}} \approx 1.6161[/tex]

[tex]\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+...}}}} \approx 2.3027[/tex]

Any references where I could read about these subjects?

Another question. Considering real [tex]x>1[/tex], we have:

[tex]\Gamma(x) - x^1 = 0[/tex] then [tex] x \approx 2[/tex]

But how does one justify that? And what are the exact values of these functions:

[tex]\Gamma(x) - x^2 = 0[/tex] then [tex] x \approx 3.562382285390898[/tex]

[tex]\Gamma(x) - x^3 = 0[/tex] then [tex] x \approx 5.036722570588711[/tex]

[tex]\Gamma(x) - x^4 = 0[/tex] then [tex] x \approx 6.464468490129385[/tex]

Thanks,

Damián.