If we define $$f_1(x)=x^\frac{1}{x}$$ $$f_{n+1}(x)=x^\frac{1}{f_n(x)}$$ Then what is the value of the following $$\lim_{k\to\infty} f_k \left( [f_k^\prime]^{-1}\left( 0\right)\right)$$ where $k\in\{\text{Odd numbers}\}$
That is to say what is the limit of the max value of $f_k$ as $k$ approached $\infty$
The first few values are as follows:
- 1.44466786100977
- 1.68566609644993
- 1.84222035704638
- 1.95368819730989
- 2.03776143885044
- 2.10376587133782
- 2.15714616594055
- 2.20131939907498
The ratio test and the root test are inconclusive.
I have no idea where to start with this, but this is basically only for fun so I may be out of my depth!