I saw this link (written in Japanese) and found an interesting problem:
Calculate $1 + \frac 2{3 + \frac 4 {5 + \cdots}}$.
The link provides the answer ($\frac 1 {\sqrt e - 1}$) and a hint that one uses Maclaurin series, but doesn't provide a detailed answer.
Can someone explain this equality? If possible, can someone also calculate
- $1^2 + \frac {2^2}{3^2 + \frac {4^2} {5^2 + \cdots}}$ and
- $1^n + \frac {2^n}{3^n + \frac {4^n} {5^n + \cdots}}$?
These two are also in the link, and answers of them are not provided and seem unsolved.
Answer of Continued fraction for $\frac{1}{e-2}$ might help.
