when does $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-...$converge

Question

when does $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}-\frac{x^7}{7!}...$ converge and diverge

where the pattern is the Thue morse sequence with adding and subtracting. $+--+-++--++-+--+...$.

I plotted it with 31 terms and found that it might diverge between 7 and 6.

I'm sure how to test it?

Answer 1

It absolutely converges, so the signs don't matter. Compare with the series for $e^x$.