For example, there are infinite series for $\pi$, $e$, $\phi$... But if I sum a finite ammount of terms, I get an approximation for the series. How do I know how much correct digits of this approximation are correct? Like, suppose I got:
$$\pi \approx\color{Blue}{3,141592}\color{Red}{4265927659}$$
How do I know that the only rigth digits for $\pi$ are $\color{Blue}{3,141592}$?
Is there a book or article about it? Is there a theorem that tells me about how much I have to calculate to get $n$ digits of correctness?