Infinite Integrations?!?!

Calculus Level 5

Let the derivative of \(F(x)\) be \(f(x)\). Therefore, in general,
\[\int f(x) dx = F(x) + c\]
However, we define a new function \(\psi\) that takes a function and returns the indefinite integral of the function with the constant as zero.Therefore,
\[\psi (f(x)) = F(x) \]

(Note that \( c = 0\) )
( The whole idea is to make that sure that in any integration during the sum the constant is assumed to be zero. I am sorry for any technical mistake while explaining the idea.)

\(\psi^{n+1}(g(x)) = \psi(\psi^{n}(g(x)))\) and \(\psi^{1}(g(x)) = \psi(g(x)) \forall n \in \mathbb{N}\)

Let \(\psi^{n}( \ln x )|_{ x =(1 + \frac{1}{n}) } = \phi(n) \)

If, \(L = \displaystyle \lim _ { n \to \infty} (n!)\) \( \phi(n) + \ln (n^e)\)

Find the value of \(e + L\)

Details and Assumptions:

  • \(\ln x\) denotes the natural logarithm of \(x\).

  • \(n!\) denotes the factorial of \(n\).

  • \(e\) is the exponential constant.


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