The Taylor Polynomial is defined as following: $P_n(x) = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \cdots + (-1)^n \dfrac{1.3.5 \cdots (2n - 3)}{2.4.6 \cdots 2n}x^n$
If $n = 4$, then the last term in the numerator expansion would be $2 \cdot 4 - 3 = 5$.
So we will have 5 terms totally, running from $x^0$ to $x^4$. But then how was the numerator generated?
First term = ?
Second term = 1
Third term = 1.3
Fourth term = 1.3.5
Fifth term = ?
I don't understand how they have $1$ for the first term as well as the fifth term. It did not make any sense to me. Any idea?
Edit
The basic function for this polynomial is $f(x) = \sqrt{x + 1}$ at $x = 0$.
Thanks,