- #1
Damidami
- 94
- 0
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.