(* General asymptotic expansion *)

m=(({{1, 0, 0}, {3, -4, 1}, {1, -2, 1}}));
f[a_,n_]=m.Transpose[{(1/(a n)),(1/(a n+1)),(1/(a n+2))}];
MatrixForm[Series[f[a,n],{n,∞,3}]]

(* Arithmetic identities, 1st pair of sets of numbers *)

(1/10)+(1/30)+(1/60)==(1/12)+(1/15)

(1/(10^2))+(1/(30^2))+(1/(60^2))==(1/(12^2))+(1/(15^2))

c3=(2/(7^3))(((1/(10^3))+(1/(30^3))+(1/(60^3)))-((1/(12^3))+(1/(15^3))))

(* Arithmetic identities, 2nd pair of sets of numbers *)

(1/16)+(1/20)+(1/240)==(1/15)+(1/24)+(1/120)

(1/(16^3))+(1/(20^3))+(1/(240^3))==(1/(15^3))+(1/(24^3))+(1/(120^3))

c2=(2/(11^2))(((1/(16^2))+(1/(20^2))+(1/(240^2)))-((1/(15^2))+(1/(24^2))+(1/(120^2))))

(* Arithmetic identities, 3rd pair of sets of numbers *)

(1/(45^2))+(1/(72^2))+(1/(144^2))+(1/(160^2))+(1/(432^2))+(1/(480^2))==(1/(48^2))+(1/(60^2))+(1/(120^2))+(1/(720^2))+(1/(1440^2))+(1/(4320^2))

(1/(45^3))+(1/(72^3))+(1/(144^3))+(1/(160^3))+(1/(432^3))+(1/(480^3))==(1/(48^3))+(1/(60^3))+(1/(120^3))+(1/(720^3))+(1/(1440^3))+(1/(4320^3))

c1=((1/45)+(1/72)+(1/144)+(1/160)+(1/432)+(1/480))-((1/48)+(1/60)+(1/120)+(1/720)+(1/1440)+(1/4320))

(* Product of involved prime factors *)

mm=2*3*5*7*11

(* Determinant *)

Det[m]

(* Computing C explicitly *)

s1={45,72,144,160,432,480};
t1={48,60,120,720,1440,4320};
s2=11{16,20,240};
t2=11{15,24,120};
s3=7{10,30,60};
t3=7{12,15};

cc=Max[Abs[{
Limit[((Plus@@((f[#,n][[1]])&/@s1))-(Plus@@((f[#,n][[1]])&/@t1))-c1/n) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[2]])&/@s1))-(Plus@@((f[#,n][[2]])&/@t1))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[3]])&/@s1))-(Plus@@((f[#,n][[3]])&/@t1))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[1]])&/@s2))-(Plus@@((f[#,n][[1]])&/@t2))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[2]])&/@s2))-(Plus@@((f[#,n][[2]])&/@t2))-c2/n^2) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[3]])&/@s2))-(Plus@@((f[#,n][[3]])&/@t2))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[1]])&/@s3))-(Plus@@((f[#,n][[1]])&/@t3))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[2]])&/@s3))-(Plus@@((f[#,n][[2]])&/@t3))) n^4,{n->∞}],
Limit[((Plus@@((f[#,n][[3]])&/@s3))-(Plus@@((f[#,n][[3]])&/@t3))-c3/n^3) n^4,{n->∞}]
}]]

N[(8833/100776960000),10]

(* Computing p explicitly *)

a=.;n=.;
kk=14;
p=N[(∑_(l=kk)^∞)(Plus@@((f[#,l^2 mm])&/@t1))+(∑_(l=kk)^∞)(Plus@@((f[#,l^2 mm])&/@t2))+(∑_(l=kk)^∞)(Plus@@((f[#,l^2 mm])&/@t3)),50]

(* Computing the box sides explicitly *)

r={(c1/((kk^2 mm+1))),(c2/((kk^2 mm+1)^2)),(c3/((kk^2 mm+1)^3))}

MatrixForm[Inverse[m].(p+(3/2)r)]

N[SingularValueList[Inverse[m]][[3]] r[[3]],50]