$\sum _{k=0}^{\infty } -\frac{2^{-2 k} j^{2 k} \Gamma \left(\frac{1}{2}-k,10\right)}{\Gamma (1+k)}+\sum _{k=0}^{\infty } -\frac{2 j^{2 k} \Gamma (-2 k) \sin (k \pi )}{\sqrt{\pi }}$

Sum[#, {k, 0, Infinity}] & /@ (Gamma[-k + 1/2, 0, 10]/k!*(j/2)^(2 k) //
    FunctionExpand)
 (*Sum[-((j^(2*k)*Gamma[1/2 - k, 10])/(2^(2*k)*Gamma[1 + k])), {k, 0, Infinity}] + 
  Sum[(-2*j^(2*k)*Gamma[-2*k]*Sin[k*Pi])/Sqrt[Pi], {k, 0, Infinity}]*)

First series we can compute from identity $\Gamma (a,z)=\int_z^{\infty } t^{a-1} \exp (-t) \, dt$

Integrate[
 Sum[-((2^(-2 k) j^(2 k) (t^(a - 1) Exp[-t] /. a -> 1/2 - k))/
   Gamma[1 + k]), {k, 0, Infinity}], {t, 10, Infinity}, Assumptions -> j >= 0]

(*-(1/2) E^(-I j) Sqrt[\[Pi]] (Erfc[(20 - I j)/(2 Sqrt[10])] + 
   E^(2 I j) Erfc[(20 + I j)/(2 Sqrt[10])])*)

Second series we can compute like this:

L = Table[
  Limit[-((2 j^(2 k) Gamma[-2 k] Sin[k \[Pi]])/Sqrt[\[Pi]]), k -> m, 
   Assumptions -> j >= 0], {m, 0, 10}]
Sum[FindSequenceFunction[L, k] // FunctionExpand, {k, 0, Infinity}]

(*Sqrt[\[Pi]] Cos[j]*)

Regards M.I.

Hi
Please execute first

Needs["PlotLegends`"];
g[j_] :=  (1/2)^0.5*Sin[Pi*0.5]/Pi*     
   NSum[Gamma[-k + 0.5, 0, 10]/k!*(j/2)^(2 k), {k, 0, 30}];
Omega = Table[g[j], {j, 0.1, 20, 0.1}];
ListLinePlot[{Omega}, PlotStyle -> Thick, PlotRange -> {-2, 2}]

to see what's going on. Note that you made a print mistake it's (j/2)^(2 k and not (k/2)^(2 k. Now I said convergence because I plotted the same function but written in terms of reduced Bessel function and found no errors.
The aim is to make the plot with k=0 to infinity.
Thanks.