We have:

Sum[(z^k Pochhammer[2, k] Pochhammer[1 + s, k])/(
 k! Pochhammer[2 + s, k]), {k, 0, Infinity}]
 (*Hypergeometric2F1[2, 1 + s, 2 + s, z]*)

then:

 Limit[(z^k Pochhammer[2, k] Pochhammer[1 + s, k])/(k! Pochhammer[2 + s, k]), s -> Infinity]
 (*(z^k Pochhammer[2, k])/k!*) 
 N[Sum[(z^k Pochhammer[2, k])/k!, {k, 0, Infinity}] /. z -> -Exp[10], 300]

 (*2.06096648272347233418959215235994959191908111757487814505583864436199\
 9184683402296197924473085819095047447295634942724479179145245736116114\
 9156752936212085037037409690812216803036585395729723051186738618366655\
 5146641627747571896005193529868156699033219139667312892465093410814678\
 504338970396392816996*10^-9*)

$$\frac{1}{\left(1+e^{10}\right)^2}$$ Symbolic value what You are looking for.