The fact that it's based on an *infinite* polynomial means that it can adjust an infinite number of times in approaching the original function. Compare the graphs (I personally like https://www.desmos.com/calculator) of

$sinx ≈ x$

$sinx ≈ x - \frac{x^3}{3!}$

$sinx ≈ x - \frac{x^3}{3!} + \frac{x^5}{5!}$

$sinx ≈ x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}$

EDIT: How do Taylor polynomials work to approximate functions? does a good job of explaining why Taylor series uses a certain series to approach the function rather than a different series.

If $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5x^5 ...$ all the way to infinity (so that the polynomial can bend an infinite number of times and approach every single point on the original function), then $f(0) = a_0$

$f'(x) = a_1 + 2a_2x + 3a_3x^2+4a_4x^3+5a_5x^4 + ...$ in which case $f'(0) = a_1$

Further, $f''(x) = 2a_2 + 6a_3x+12a_4x^2 + 20a_5x^3 + ...$ in which case $f''(0) = 2a_2$

$f'''(x) = 6a_3 + 24a_4x+ 60 a_5x^2+...$ in which case $f'''(0) = 6a_3$

$f''''(x) = 24a_4 + 120a_5x + ...$ in which case $f''''(0) = 24a_4$

Now we have a pattern for all of the coefficients $a_n = \frac{f^n(0)}{n!}$ that we can plug back into the original $f(x)$