Let $p_n/q_n$ be the $n$-th convergent of the continued fraction expansion of a real number $\alpha$.
It is known that $|p_n-q_n\alpha|$ is very small tending to $0$ as $n$ tends to infinity.
In this paper we establish a method how to express $p_n-q_n\alpha$ in terms of integrals when $\alpha$ is an $e$-type real number and its continued fraction expansion is quasi-periodic.
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