Manipulate[Grid[{{"Terminos de la serie"},{If[mostrar,Style[Take[Table[(("-1"^k)/(Factorial[k]*Factorial[("n"+k)]))*((Defer["x"/2])^(2*k+"n")),{k,0,10}],l+1],FontSize->24],""]},
{"Serie semi desarrollada"},{Sum[((((-1))^i)/(i!(i+n)!))((((("x")/2)))^(2i+n)),{i,0,l,1}]},
{"Suma de la serie"},{Sum[((((-1))^i)/(i!(i+n)!))*((((x/2)))^(2i+n)),{i,0,l}]},
{Sum[(((-1)^i)/(Factorial[i]*Factorial[(n+i)]))*((x/2)^(2*i+n)),{i,0,l}]},
{"Valor de BesselJ"},{BesselJ[n,x]},
{"Error"},{BesselJ[n,x]-(Sum[((((-1))^i)/(i!(i+n)!))((((x/2)))^(2i+n)),{i,0,l}])},
{Switch[control,1,Plot[{BesselJ[n,t]},{t,0,r+x},ImageSize->Large],2,Plot[Sum[(((-1)^i)/(i!*(i+0)!))*((((t/2)))^(2i+0)),{i,0,20,1}],{t,0,r+x},ImageSize->Large],3,Show[{Plot[{BesselJ[n,t]},{t,0,r+x}],Plot[Sum[(((-1)^i)/(i!*(i+0)!))*((((t/2)))^(2i+0)),{i,0,20,1}],{t,0,r+x},PlotStyle->Orange]},ImageSize->Large]]}}],{{n,0,"Número de la función de Bessel"},0,3,1},{{l,0,"Terminos de la serie"},0,10,1},{{x,.1,"Valor de x"},.1,10,.1},{{r,.1,"Rango de la función"},.1,20,.1},{{mostrar,True,"Mostrar Terminos de la serie"},{True,False}},{{control,1},1,3,1}]

Trabajo realizado con el apoyo del Programa UNAM-DGAPA-PAPIME PE110517