I am not dealing with mathematics on a deep basis, but when trying to get $\pi$ from "Leibniz" series, I find that
4 - (series) approximates $\pi$ as well as 4 * series, under condition that there are not many terms present.
For example:
$$ 4 - \left( 1 + \frac {1}{3} - \frac {1}{5} + \frac {1}{7} - \frac {1}{9}\right) \approx 4 * \left(1 + \frac {1}{3} - \frac {1}{5} + \frac {1}{7} - \frac {1}{9}\right) $$ How would you describe/call this? I am asking for the name of the phenomenon.
Nothing too special. $\pi$ is rather close to $3.2=\frac{16}5$, which happens to be the solution of the equation $4-\frac x4=x$.
($4\times$)The partial sums of the Leibniz series calculate numbers $\pi_n$ which get closer and closer to $\pi\approx 3.1416$ as $n$ grows. Likewise, $u_n=4-\frac{\pi_n}4$ get closer and closer to $4-\frac{\pi}4\approx3.2146$.
Now, it's basically a matter of the first terms of $\pi_n$ being closed-but-not-too-closed to $\pi$, the first terms of $u_n$ being closed-but-not-too-closed to $4-\frac\pi4$, and $\pi$ being closed-but-not-too-closed to $4-\frac\pi4$. However, this proximity of estimates eventually disappears, as it ought to.