Size[poly_,var_]:=-Series[poly,{var,0,Max[Exponent[poly,var],1]}][[4]]
CoeffList[poly_,var_,terms_]:=PadRight[Series[poly,{var,0,Exponent[poly,var]+terms+1}][[3]],Exponent[poly,var]+terms+1]
FunctionDivAlg[polya_,polyb_,var_]:=Module[{aSize,bSize,SizeDif, output,q,r,aCoeffs, bCoeffs,bMatrix,aVector,cVector,ai,ci},
aSize=Size[polya,var];
bSize=Size[polyb,var];
SizeDif=aSize-bSize;
If[SizeDif<0, output={0,polya},
aCoeffs=CoeffList[polya,var,SizeDif];
bCoeffs=CoeffList[polyb,var,SizeDif];
bMatrix=Table[Table[If[i>0,bCoeffs[[i]],0],{i,1+j,1+j-SizeDif,-1}],{j,0,SizeDif}];
aVector=Table[{aCoeffs[[j]]},{j,1,SizeDif+1}];
cVector=Table[Table[ci[j],{i,1,1}],{j,SizeDif,0,-1}];
cVector=cVector/.ToRules[Reduce[aVector==bMatrix.cVector,ci]];
q=Sum[cVector[[i+1]]*x^(i-SizeDif),{i,0,SizeDif}][[1]];
r=Expand[polya-polyb*q];
{q,r},{0,polya}]]
EgyptianConvergents[QList_]:=Simplify[Table[Sum[(-1)^(i+1)/QList[[i]],{i,1,j}],{j,1,Length[QList]}]]

{q1,r1}=FunctionDivAlg[ x^3+8x^2+5x+4,4x^2-6x+11,x]

{q2,r2}=FunctionDivAlg[(x^3+8x^2+5x+4)*(q1),r1,x]

{q3,r3}=FunctionDivAlg[(x^3+8x^2+5x+4)*(q1)*(q2),r2,x]

{q4,r4}=FunctionDivAlg[(x^3+8x^2+5x+4)*(q1)*(q2)*(q3), r3,x]

Simplify[1/q1-1/q2+1/q3-1/q4]

EgyptianConvergents[{q1,q2,q3,q4}]

qseries={q1,q2,q3,q4};
Manipulate[Plot[{Sum[(-1)^(j+1)/qseries[[j]],{j,1,Length[qseries]}],Sum[(-1)^(j+1)/qseries[[j]],{j,1,i}]},{x,-10,10}],{i,1,Length[qseries],1}]

Sum[(-1)^(i+1)/qseries[[j]],{j,1,Length[qseries]}]