Asymptotic Expansion: an illustration

Example 1.

Clear[All]

F[x_]:= (∫_x^∞)(1/t)(E^(x-t ))\[DifferentialD]t

N[F[1]]

0.596347362323194`

N[F[2]]

N[F[10]]

0.091563333939788060.0915633

S[x_,n_]:= Sum[(((((-1))^m)*m!)/(x^(m+1))),{m,0,n}]

S[10, 5]

Do[PS[i]=N[S[10,i]],{i,1,30}];

Table[PS[i],{i,1,27}]

ListPlot[%85]

Do[PSN[i]=N[S[10,i]]-N[F[10]],{i,1,30}];

Table[PSN[i],{i,1,27}]

ListPlot[%90]

Do[PS[i]=N[S[20,i]],{i,1,40}];Table[PS[i],{i,1,35}]

ListPlot[%5]

N[F[30]]

N[S[10,30]]

N[S[10,31]]

N[S[10,40]]

Do[P[i]=N[S[30,i]],{i,1,60}]

Table[P[i], {i,1,55}];

ListPlot[%]

N[F[30]]

Do[Q[i]=N[S[30,i]]-0.03228973875898009,{i,1,60}]

Table[Q[i], {i,1,55}];

ListPlot[%]

Example 2.

G[g_]:= (∫_(-∞)^∞)(E^(- (x^2)-  g  (x^4)))\[DifferentialD]x;

N[G[0.1]]

N[G[0.5]]

G[g]=  (∫_(-∞)^∞)(E^(- (x^2)))(1 - g  (x^4) + ((g^2)/2) (x^8)  + ... )  \[DifferentialD]x

(∫_(-∞)^∞)(E^(- (x^2)))(x^4)  \[DifferentialD]x

(∫_(-∞)^∞)(E^(- (x^2)))(x^8)  \[DifferentialD]x

H[g_,n_]:= (((g^n) (((-1))^(n )))/(n!))(∫_(-∞)^∞)(E^(- (x^2)))(x^(4  n ))  \[DifferentialD]x;

H[0.5,5]

Do[R[i]=N[H[0.5,i]],{i,1,10}]

Table[R[i], {i,1,6}];

ListPlot[%]

Do[R[i]=N[H[0.5,i]-1.4863108176097255`],{i,1,10}]

Table[R[i], {i,1,6}];

ListPlot[%]