Can Wynn's $\epsilon$ algorithm be used for sequence limit?

Question

Let's assume we have a sequence $(a_n)$, which converges to some limit $L = \lim_{n\to\infty} a_n$. However, we are able to calculate only first $N$ terms of the sequence. It is clear that, in general, $L \neq a_N$. Is it possible, however, to use some series accelerator algorithm, e.g. the Wynn epsilon method on the sequence $\{a_1, a_2,\ldots a_N\}$ to obtain a new sequence $(b_n)$ so that the final element of the new sequence, $b_N$ is a better approximation to $L$ than $a_N$, i.e. $|b_N - L|< |a_N - L|$? Then by repeating the procedure, we can arrive as close to $L$ as is necessary.