Why Maclaurin Series of $\tan(x)$ consists of only positive coefficients, that would mean $\tan(x)$ always has a positive value for any positive real number, but it is clear from the graph of $\tan(x)$ that this is not true.
Maclaurin Series of $\text{tan}(x)$
taylor expansion
Best Answer
This is because the formula for the Taylor/MacLaurin series is only true around a certain point; and is only defined for sections of a function which are smooth around this point. $\tan(x)$ has many asymptotes, so is discontinuous in these points, so is not derivable in these points, so is not smooth in these points.
To put it simply, your Taylor Series is for $\tan(x)$ defines $\tan(x)$ for $x \in ]-\frac{\pi}{2}, \frac{\pi}{2}[$. And for $x \in [0, \frac{\pi}{2}[$, $x$ is indeed positive. You can use the formula to find approximations around other smooth sections, etc.