Taylor Series of $sinh(x)$ and $cosh(x)$
We can derive the Taylor Series for $sinh(x)$ and $cosh(x)$ as follows:
$$
\begin{aligned}
sinh(x) & = {{e^x - e^{-x}} \over 2}\\
cosh(x) & = {{e^x + e^{-x}} \over 2}\\
sinh^{(1)}(x) & = {{e^x + e^{-x}} \over 2} = cosh(x)\\
cosh^{(1)}(x) & = {{e^x - e^{-x}} \over 2} = sinh(x)\\
sinh(0) & = 0\\
cosh(0) & = 1\\
sinh^{(n)}(0) & = 0 \, \mbox{when n is even}\\
& = 1 \, \mbox{when n is uneven}\\
cosh^{(n)}(0) & = 1 \, \mbox{when n is even}\\
& = 0 \, \mbox{when n is uneven}\\
sinh(x) & = 0 + 1 \cdot x + 0 \cdot {{x^2} \over 2!} + 1 \cdot {{x^3} \over 3!} + 0 \cdot {{x^4} \over 4!} + 1 \cdot {{x^5} \over 5!} + .....\\
& = x + {{x^3} \over 3!} +{{x^5} \over 5!} +{{x^7} \over 7!} +...\\
& = \sum_{j=0}^\infty {{x^{2j+1}} \over (2j+1)!}\\
cosh(x) & = 1 + 0 \cdot x + 1 \cdot {{x^2} \over 2!} + 0 \cdot {{x^3} \over 3!} + 1 \cdot {{x^4} \over 4!} + 0 \cdot {{x^5} \over 5!} + .....\\
& = 1 + {{x^2} \over 2!} +{{x^4} \over 4!} +{{x^6} \over 6!} +...\\
& = \sum_{j=0}^\infty {{x^{2j}} \over (2j)!}\\
\end{aligned}
$$
Taylor Series for $arctanh(x)$
The derivatives of $arctanh(x)$ can be determined and therefore we can deduce a Taylor Series for it:
$$
\begin{aligned}
arctanh^{(1)}(x) & = {1 \over {1-x^2}}\\
arctanh^{(2)}(x) & = {{2x} \over {(1-x^2)^2}}\\
arctanh^{(3)}(x) & = {{2} \over {(1-x^2)^2}} + {{8x^2} \over {(1-x^2)^3}}\\
arctanh^{(4)}(x) & = {{24x} \over {(1-x^2)^3}} + {{48x^3} \over {(1-x^2)^4}}\\
arctanh^{(5)}(x) & = {{24} \over {(1-x^2)^3}} + {{288x^2} \over {(1-x^2)^4}} + {{384x^4} \over {(1-x^2)^5}} \\
arctanh^{(6)}(x) & = {{720x} \over {(1-x^2)^4}} + {{3840x^3} \over {(1-x^2)^5}} + {{3840x^5} \over {(1-x^2)^6}} \\
arctanh^{(7)}(x) & = {{720} \over {(1-x^2)^4}} + {{17280x^2} \over {(1-x^2)^5}} + {{57600x^4} \over {(1-x^2)^6}} + {{46080x^6} \over {(1-x^2)^7}} \\
\end{aligned}
$$
Getting the values for these derivatives at zero:
$$
\begin{aligned}
arctanh(0) & = 0\\
arctanh^{(1)}(0) & = {1 \over {1-0^2}}\\
& = 1\\
arctanh^{(2)}(0) & = {{2 \cdot 0} \over {(1-0^2)^2}}\\
& = 0\\
arctanh^{(3)}(0) & = {{2} \over {(1-0^2)^2}} + {{8 \cdot 0^2} \over {(1-0^2)^3}}\\
& = 2\\
arctanh^{(4)}(0) & = {{24 \cdot 0} \over {(1-0^2)^3}} + {{48 \cdot 0^3} \over {(1-0^2)^4}}\\
& = 0\\
arctanh^{(5)}(0) & = {{24} \over {(1-0^2)^3}} + {{288 \cdot 0^2} \over {(1-0^2)^4}} + {{384 \cdot 0^4} \over {(1-0^2)^5}} \\
& = 24\\
arctanh^{(6)}(0) & = {{720 \cdot 0} \over {(1-0^2)^4}} + {{3840 \cdot 0^3} \over {(1-0^2)^5}} + {{3840 \cdot 0^5} \over {(1-0^2)^6}} \\
& = 0\\
arctanh^{(7)}(0) & = {{720} \over {(1-0^2)^4}} + {{17280 \cdot 0^2} \over {(1-0^2)^5}} + {{57600 \cdot 0^4} \over {(1-0^2)^6}} + {{46080 \cdot 0^6} \over {(1-0^2)^7}} \\
& = 720\\
arctanh^{(n)}(0) & = 0 \, \mbox{when n is even}\\
& = (n-1)! \, \mbox{when n is uneven}\\
\end{aligned}
$$
Putting into the Taylor Expansion:
$$
\begin{aligned}
arctanh(x) & = 0 + x + 0 + 2! \cdot {{x^3} \over 3!} + 0 + 4! \cdot {{x^5} \over 5!} + 0 + 6! \cdot {{x^7} \over 7!} + 0 + ....\\
& = x + {{x^3} \over 3} + {{x^5} \over 5} + {{x^7} \over 7} + {{x^9} \over 9} + ...\\
& = \sum_{j=0}^n {{x^{2j+1}} \over {2j+1}}
\end{aligned}
$$