Suppose we have some analytic $f(z)$, with Taylor expansion at $z=0$:
$$f(z) = \sum_{n=0}^{\infty} c_n z^n$$
Now, suppose we want to approximate the $N$'th partial sum, but without literally summing all $N$ terms.
Of course, the best way is if there exists a closed-form expression for the "indefinite sum" of your series. If that doesn't exist, then there are several ways to do this numerically. One is to express the partial sum as a difference of two sums:
$$f_N(z) = \sum_{n=0}^{N} c_n z^n = \left(\sum_{n=0}^{\infty} c_n z^n\right) - \left(\sum_{n=0}^{\infty} c_{n+N+1} z^{n+N+1}\right)$$
Series acceleration methods (such as Shanks, Levin's, etc) can then be applied to both terms separately. As an example, using Wynn's epsilon on both series separately yields an approximated partial sum as a difference of two Pade approximants.
We may also be able to express the partial sum as a single infinite sum, if there are no convergence issues:
$$ f_N(z) = \sum_{n=0}^{N} c_n z^n = \left(\sum_{n=0}^{\infty} c_n z^n - c_{n+N+1} z^{n+N+1}\right) $$
and then try using series acceleration methods on that, which sometimes converges slightly faster than the last one.
Both of these methods can give quite good results.
However, the problem in my situation is that sometimes there isn't enough cancellation on the higher-order terms (i.e., $> N$). That is, due to the way that Pade approximants (and rational approximations in general) work, it is typically the lower-order terms that are as accurate as possible. In my situation, I would be happiest if I had stronger cancellation on the higher-order terms, even if the coefficients on the lower order terms aren't exactly right.
One thing I have tried is multiplying the rational approximant by a "correction term" that attempts to cancel out as many higher-order terms as possible. For instance, if the series acceleration methods yield a rational approximation
$$\frac{a_0 + a_1 z + a_2 z^2 + ... + a_A z^A}{1 + b_1 z + b_2 z^2 + ... + b_B z^B}$$
Then you can rewrite the denominator as
$$\frac{a_0 + a_1 z + a_2 z^2 + ... + a_A z^A}{1 - D}$$
where $D = -(b_1 z + b_2 z^2 + ... + b_B z^B)$, so that this looks like a geometric series. You can then multiply by $(1-(1-b_1 z + b_2 z^2 + ... + b_B z^B)^M)$, for some large $M$, which will cause all higher-order terms to cancel beyond a certain point. This typically destroys the approximation entirely, though.
However, it seems like it should be possible to do this in a way such that, once you multiply a correction term similarly to the above, the net result remains a good approximation.
My question: is there any better way to do this, or general way to approximate these partial sums in general?