Is there a way to accelerate the convergence of alternating Taylor's series such as that $$ \sum_{n=0}^\infty (-1)^n a_n x^n=A $$ approaches its value with less number of terms.
Since its an alternating series, I wondered if Euler's transformation might do the trick where $$ \sum_{n=0}^\infty (-1)^n a_n =\sum_{n=0}^\infty (-1)^n \frac{(\Delta^n a)_0}{2^{2n+1}} $$ Where $$ (\Delta^n a)_0 = \sum_{k=0}^n (-1)^k \binom{n}{k} a_{n-k} $$ If anyone knows any better way to accelerate the convergence of Taylor's series, please let me know. Thank you in advance.