# Integration is not correct for the expression with a fraction?

What value Px2[16, 4] or N[Px22[16, 4]] is correct?

ClearAll["Global*"]
y = 5;
Psi[r_, n_] = (2 E^(-(r/n)) Sqrt[n!/(-1 + n)!] Hypergeometric1F1[
1 - n, 2, (2 r)/n])/n^2;

VB2[r_] := Exp[-r*N[(154419325000 Sqrt[25034/57057])/68663001981]];
VB22[r_] := Exp[-r*(154419325000 Sqrt[25034/57057])/68663001981];

Px2[n1_, n2_] :=
Integrate[Psi[r, n2]*VB2[r]*Psi[r, n1]*r^2, {r, 0, \[Infinity]}];
Px22[n1_, n2_] :=
Integrate[Psi[r, n2]*VB22[r]*Psi[r, n1]*r^2, {r, 0, \[Infinity]}];

In[742]:= Px2[16, 4]

Out[742]= 0.000266864

In[743]:= N[Px22[16, 4]]

During evaluation of In[743]:= General::munfl: 1/2.25932*10^16^21 is too small to represent as a normalized machine number; precision may be lost.

Out[743]= 0.


• Try: VB2[r_] := Exp[-r*N[(154419325000 Sqrt[25034/57057])/68663001981, 20]]; and N[Px22[16, 4], 20] ? Jan 29, 2023 at 12:49
• Read "Machine-Precision Numbers" from tutorial/Numbers#31699 in the help. Calculations with Machine numbers are fast but not exact. Jan 29, 2023 at 13:26

The results are the same and you can prove it.

With

y = 5;
Psi[r_, n_] = (2 E^(-(r/n)) Sqrt[n!/(-1 + n)!] Hypergeometric1F1[
1 - n, 2, (2 r)/n])/n^2;
VB2[r_] := Exp[-r*N[(154419325000 Sqrt[25034/57057])/68663001981]];
VB22[r_] := Exp[-r*(154419325000 Sqrt[25034/57057])/68663001981];


Then, we consider the two integrals. Note the use of Rationalize

Px2[n1_, n2_] :=
Integrate[
Rationalize[Psi[r, n2]*VB2[r]*Psi[r, n1]*r^2, 0], {r, 0, Infinity}];
Px22[n1_, n2_] :=
Integrate[
Rationalize[Psi[r, n2]*VB22[r]*Psi[r, n1]*r^2, 0], {r, 0,
Infinity}];


Now, you can check their ratio

Px2[16, 4]/Px22[16, 4] // N


1.

If you want to be more precise about the statement

Px2[16, 4]/Px22[16, 4] // N[#, 100] &


0.99999999999999997884679787110503763789829242808078911831345030580990\ 159960929487143929995821867831721`