The Taylor series is: x + (x^3)/3! + (x^5)/5! + (x^8)/8! = ....

In terms of sigma notation it is: sigma(x^(2k+1)/(2k+1)!) as k goes from 0 to infinity.

To figure it out, recall that sinh(x) = (exp(x)-exp(‑x))/2, so you can

get the series for exp(x) and the series for exp(-x), subtract the latter from the former, then divide by 2. If this isn't detailed enough, let me know.