What is the area of the region bounded by $y = f_n(x)$ and the $x$-axis as $n$ gets large?
$f_0=1-|x|$
$f_n=1-|1-2f_{n-1}(x)|$
Assume the domain of $f$ to be the real numbers.
By trying various values for $n$ and $x$ I got that $f_n(x) = 2^{n}\cdot f_0(x)$
By integrating this I got that the area is given by $2^{n-1}\cdot x\cdot (2-|x|)$.
However, I was also told that the area is in fact $1$ as $n$ approaches infinity.
The formula I came up with (guessed) seems to check out if I integrate $2^{n-1}\cdot f_0(x)$ between certain values and for a certain $n$ and compare it to manually expanding that certain $f_n$ and integrating it between the same values.
My questions are:
- Is my formula wrong?
- How can we prove the area is 1?