I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting something serious .Recently, I was cleaning my room where I found my old diary in which I had scribbled down something odd , which I don't remember solving , and still am wondering ,how to?
I would like to reproduce the question in more modern notation as- Lets say $$e^ \pi = [23; a_1, a_2, a_3, \ldots ]$$ and $$\pi ^ e = [22; b_1, b_2, b_3, \ldots]$$
Then compute the values of $$I_1=\sum_{n=1}^{\infty}\frac{1}{a_1^2+a_2^2+ \cdots a_n^{2}}$$ and $$I_2=\sum_{n=1}^{\infty}\frac{1}{a_1b_1+a_2b_2+\cdots a_nb_n}$$
Just some thoughts:
For $I_1$;
since the a's in
$$I_1=\sum_{n=1}^{\infty}\frac{1}{a_1^2+a_2^2+ \cdots a_n^{2}}$$
do not have a pattern or any relationship I know of, using analytical techniques to sum or accelerate the sum are beyond me and may not even exist. I did some numerical work and hope I am close. The idea is to truncate the series at something less than infinity until there seems to be some convergence,
$$I_1=\sum_{n=1}^{159999}\frac{1}{a_1^2+a_2^2+ \cdots a_n^{2}} \ \approx .050668$$
I would offer as a conjecture that the first 5 of the given digits are correct?
Although the terms are getting smaller, convergence is quite slow and that is the limit of what I can do for the reasons already given. The small number of digits makes it impossible to try to guess at a closed form using integer relation algorithms or PSLQ.
Update:
An improvement in design allowed for a faster algorithm with it a higher upper index.
$$I_1=\sum_{n=1}^{2000000}\frac{1}{a_1^2+a_2^2+ \cdots a_n^{2}} \ \approx .0506703$$
The conjecture about the first 5 digits is now iffy and some insight is needed...