Defining Functions: Different Ways

f[x_]:= Sin[x]
f1=Sin
f2=Sin[#]&

Clear[x,g,g1,g2,dg]
g[x_]:= Sin[(x^2)]
g2=Sin[((#)^2)]&
g1[x_]:= Module[
{y},
y=(x^2);
Sin[y]]
dg[x_]=D[g[x],x]

Power Series: Automagically

Series[
BesselJ[1,x],
{x, 0,13}]
Series[
(E^x),
{x, 0,13}]

Power Series: Recurssion for Bessel

Clear[a]
p=1;
a[0]=0;
a[1]=1/2;
a[n_]:=((a[n-2])/((p^2)- (n^2)))
Table[ a[n], {n, 0, 12}]

dg[1.2]

Plot[g3[x],{x, 0, 12}]

Power Series: Automatic
Recurssion for Bessel

Inspired by a notebook someone sent me to fix last night.  We can build the recurrsion automatically. I included one extra

Clear[y,x,a,n]
iMax=12;
y[x_]:=Sum[a[i] (x^(i+δ)),{i,0, iMax}]
BesselEq =(x^2)y''[x]+x y'[x]+((x^2)-(n^2))y[x];
Simplify[
CoefficientList[BesselEq,x]
]//TableForm