Integral involving up-arrow notation

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Discussion Overview

The discussion revolves around the evaluation of an integral involving the sine function raised to a power tower using up-arrow notation. Participants explore the implications of this notation in the context of polar graphs and the convergence of the integral as parameters change. The scope includes mathematical reasoning and conceptual clarification regarding convergence criteria and the behavior of the function.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the integral $$\frac {1} {2} \int_0^\pi (sin(\theta)\uparrow \uparrow \infty)^{2} d\theta$$ may be meaningless due to the nature of the infinitely tall power tower.
  • Another participant references convergence criteria for infinite power towers, stating that convergence occurs if $$e^{-e} \leq z \leq e^{1/e}$$ and suggests that since $$\sin(0) < e^{-e}$$, the integral diverges.
  • Some participants discuss the oscillatory behavior of the function $$r(\theta)=\sin(\theta)\uparrow\uparrow k$$ as $$k$$ varies, particularly noting that it oscillates between 0 and 1 at the endpoints of the interval.
  • There is a suggestion that the limit of the integral as $$k \to \infty$$ may not be meaningful since $$k$$ is not a continuous parameter.
  • One participant proposes using the Monotone Convergence theorem to analyze the convergence of the power tower, questioning which values of $$k$$ yield monotonic iterations.
  • Participants express uncertainty about how to accurately perform iterations in Wolfram and the implications of using up-arrow notation in this context.
  • Links to external resources are shared to provide additional context on power towers and convergence.

Areas of Agreement / Disagreement

Participants express differing views on the meaningfulness of the integral and the convergence of the power tower. There is no consensus on whether the limit of the integral as $$k$$ approaches infinity can be evaluated or if it is well-defined.

Contextual Notes

There are limitations regarding the assumptions made about the convergence of the power tower and the behavior of the sine function at specific points. The discussion highlights the complexity of defining limits and convergence in this context.

  • #31
The graph posted in #15 is misleading at it's endpoints outside of ##(e^{-e},e^{1/e})##: The command used to plot it was:
[CODE title="Mathematica"]Nest[Sin[theta]^# &, 1, 800]
[/CODE]
So that's just iterating the tetration of ##\sin(t)## 800 times. But when ##\sin(t)<e^{1/e}##, the tetration is no longer single valued but rather enters a 2-cycle oscillation and selecting the even number of iterations was just selecting the higher value. Consider the output:
[CODE title="Mathematica"]In[59]:= Nest[Sin[0.02]^# &, 1, 500]
Nest[Sin[0.02]^# &, 1, 501]

Out[59]= 0.884202

Out[60]= 0.0314587[/CODE]

This is what a plot of ##\sin(t)\uparrow\uparrow \infty ## really looks like on ##(0,\pi)##:
tetrationbifurcation.jpg
 
Last edited:
  • Informative
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