AHSrearranged[L_,step_:100]:=Module[{k=0,sum=0,even=1,odd=2},Do[If[sum≤L,
sum=sum+1/even;k=k+1;Print["φ(",k,") = ",even,"\t+",1/even,"\t",sum," = ",N[sum]];even=even+2;,
sum=sum-1/odd;k=k+1;Print["φ(",k,") = ",odd,"\t-",1/odd,"\t",sum," = ",N[sum]];odd=odd+2;],{step}]]

AHSrearrangedG[L_,step_:20]:=(even=1;odd=2;evenc[0]={};oddc[0]={};sum[0]=0;Do[If[sum[j]≤L,
sum[j+1]=sum[j]+1/even;evenc[j+1]=Append[evenc[j],{even,0}];oddc[j+1]=oddc[j];even=even+2,
sum[j+1]=sum[j]-1/odd;oddc[j+1]=Append[oddc[j],{odd,0}];evenc[j+1]=evenc[j];odd=odd+2];Print[ListPlot[Table[sum[l],{l,1,j+1}],PlotRange->{{1,2 step},{-1,2}},Axes->True,PlotStyle->{Green,AbsolutePointSize[5]},AxesOrigin->{1,0},ImageSize->800,Epilog->{{{Green,Line[{{0,L},{2step,L}}]},Blue,AbsolutePointSize[5],Point[oddc[j+1]]},{Red,AbsolutePointSize[5],Point[evenc[j+1]]}}]];,{j,0,step,1}])

AHSrearranged[5/4,30]

AHSrearrangedG[5/4,100]