I have a function $y=\frac{1}{x}$. If $x = 1$, then $y = 1$; if $x = 2$, then $y = 0.5$, etc. If I wanted to add the previous term to the current term, I would have $x = 1.5$, $x = 2.167$ etc.
The terms in the sequence could be represented as:
Term 1:$\ x+\frac{1}{x}$
Term 2:$\ x+\frac{1}{x+\frac{1}{x}}+\frac{1}{x}$
Term 3:$\ x+\frac{1}{x+\frac{1}{x}}+\frac{1}{x+\frac{1}{x+\frac{1}{x}}+\frac{1}{x}}+\frac{1}{x}$
If the initial value was set to $x = 1$, the sequence would continue $x = 2$, $x = 2.5$, $x = 2.9$, with the recurrence relationship $a_n =a_{n-1} +\frac{1}{a_{n-1} }$ .
When I plot the first $1000$ numbers in the sequence, they converge to fit a power law line of best fit$\ 1.4234x^{0.4993}$, but I have no idea why. Shown below is the plot of the first $1000$ numbers and the line of best fit:
Does a closed form representation of this sequence exist? Something like a function of the Fibonacci numbers, or a sum of series, or a sum of series within a sum of series? Many thanks!