h[x_,k_] = Sum[(-1)^i*x^(2*i+1)/(2*i+1)!,{i,0,k}];

h2[x_,k_] = Series[H[x],{x,0,k}];

Manipulate[
Show[
Plot[h[x,k],{x,-4*π,4*π},PlotRange->{{-4*π,4*π},{-1.5,1.5}},PlotStyle->Red],
Plot[Sin[x],{x,-4*π,4*π},PlotRange->{{-4*π,4*π},{-1.5,1.5}},PlotStyle->Black]
],
{k,0,20,1,Appearance->"Labeled"}]

f2[k_] = Product[3*i-1,{i,1,k}];

f0[k_]= Product[3*i,{i,1,k}];

g[x_,k_] = 1+Sum[x^(3*i)/(f0[i]*f2[i]),{i,1,k}];
G[x_] = 1+Sum[x^(3*i)/(f0[i]*f2[i]),{i,1,Infinity}];

Manipulate[
Show[
Plot[g[x,k],{x,-12,4},PlotRange->{{-12,4},{-2,4}},PlotStyle->Red],
Plot[G[x],{x,-12,4},PlotRange->{{-12,4},{-2,4}},PlotStyle->Black]
],
{k,0,40,1,Appearance->"Labeled"}]