Is this not calculus? (2)

For the integral, We could use the following method:

$\displaystyle \int_a^b f(x) \ dx + \int_{f(a)}^{f(b)} f^{-1}(x) \ dx = bf(b) - af(a)$ . For $a = e^{-e}$ and $b = e^{1/e}$ , $f(a) = \frac{1}{e}$ and $f(b) = e$ .

$\displaystyle \int_{1/e}^e x^{1/x} \ dx \approx 2.65861 \ \implies \left( e^{1/e} \right) e - \left( e^{-e} \right) \frac{1}{e} - 2.65861 \approx 1.24413...$

OR

we could attempt to approximate the integral (which has an extraordinarily slow rate of convergence!) by taking a Riemann sum with very small step values:

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terms = []
x = 1.4446678610                                         # upper bound for the integral
height = 1300000                                         # height of the power tower (very good approximation)
x_0 = x

while x >= 1:
    for i in range(height):
        x = pow(x_0, x)
    terms.append(x*0.0001)
    x_0 -= 0.0001                                        # lower bound for the integral
    if x_0 < 0.06598803584531:
        break
print(sum(terms)) 

After finding $A$ , pass it through the following program several times (with a huge "height") to gauge its convergence.

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A = float(input("Enter new A: "))
height = 1300000
x_0 = A
for i in range(height):
    A = pow(x_0, A)
print(A)

We see that $f(A)$ comes out to about $1.34007$ , and $f(f(A)) \approx 1.59505$ . The power tower does not converge for values greater than $e^{1/e} \approx 1.444678610$ . Checking $f(f(f(A))) \approx f(1.59505)$ , we receive an error if we use a height that is too large. In fact, a height of $7$ prints an answer of $4.6687015696911937e+30$ , a clear indicator that $f(f(f(A)))$ diverges.

Interestingly, it appears $f$ crawls (rather slowly) to its convergent values, but explodes to infinity immediately after $x$ becomes greater than the upper bound of the domain of $f$ .