Integral involving up-arrow notation

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SUMMARY

The discussion centers on evaluating the integral $$\frac {1} {2} \int_0^\pi (\sin(\theta) \uparrow \uparrow \infty)^{2} d\theta$$, which involves the concept of up-arrow notation for infinite power towers. Participants conclude that the integral diverges because the infinite power tower converges only when the base is within the interval $$[e^{-e}, e^{1/e}]$$, and since $$\sin(0) < e^{-e}$$, the integral does not converge. The conversation also touches on the limitations of up-arrow notation and the implications for polar graphs.

PREREQUISITES
  • Understanding of up-arrow notation and its implications for infinite power towers.
  • Familiarity with polar coordinates and the formula for the area under a polar curve.
  • Knowledge of convergence criteria for sequences and series, particularly in relation to power towers.
  • Basic calculus skills, including integration techniques and limits.
NEXT STEPS
  • Research the convergence criteria for infinite power towers, specifically the conditions $$e^{-e} \leq z \leq e^{1/e}$$.
  • Study the application of the Monotone Convergence Theorem in the context of sequences and integrals.
  • Explore the use of Lambert's W function in solving equations of the form $$y = x^y$$.
  • Investigate the implications of Riemann and Lebesgue integrability concerning unbounded functions.
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced topics like infinite series, polar coordinates, and the convergence of power towers.

  • #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
 
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