(* Series from Enting, J Phys. A 27 (1994) 6987 *)
(* cn contains even terms only starting at order 4 *)

cn={16,72,288,1200,5376,25480,125504,634608,3269680,17086168,90282240,481347152,2585485504,13974825960,75941188736,414593263952,2272626444528,12502223573304,68996534259040,381858968527680,2118806030647328,11783826597027256,65674579024955904,366728645195006000,2051443799934043632,11494250259278105304,64499139095733378176,362436080938852037648,2039249170926323834880,11487673072269872540904,64786142191741932873984,365754067103461706996304,2066925549185792626090544,11691314122170272566638200,66188283453887221177721568,375021938737150106426702208,2126523853550658555941372768};

Rats=Sqrt[Drop[cn,-1]/Drop[cn,1]]//N

xc=Sqrt[2]-1//N (* Exact xc *)

exactxc=Table[xc,{n,Length[Rats]}];

ListPlot[{Rats,exactxc},PlotRange->{0.4,0.51}]

dnRats=Table[n*Rats[[n]]-(n-1)*Rats[[n-1]],{n,2,Length[Rats]}]//N

ListPlot[{Rats,exactxc,dnRats},PlotRange->{0.4,0.51}]

(* All even terms starting from order zero *)

cn={0,0,16,72,288,1200,5376,25480,125504,634608,3269680,17086168,90282240,481347152,2585485504,13974825960,75941188736,414593263952,2272626444528,12502223573304,68996534259040,381858968527680,2118806030647328,11783826597027256,65674579024955904,366728645195006000,2051443799934043632,11494250259278105304,64499139095733378176,362436080938852037648,2039249170926323834880,11487673072269872540904,64786142191741932873984,365754067103461706996304,2066925549185792626090544,11691314122170272566638200,66188283453887221177721568,375021938737150106426702208,2126523853550658555941372768};
chin={0,0,1,8,60,416,2791,18296,118016,752008,4746341,29727472,185016612,1145415208,7059265827,43338407712,265168691392,1617656173824,9842665771649,59748291677832,361933688520940,2188328005246304,13208464812265559,79600379336505560,479025509574159232,2878946431929191656,17281629934637476365,103621922312364296112,620682823263814178484,3714244852389988540072,22206617664989885664363,132657236460768679560864,791843294876287279547520,4723112509660327575046688,28152514246598001579534217,167696255471026758161692328,998303936498277539688401212,5939502715888619728011515904,35318214476286590871820680287};
magn={1,0,-2,-8,-34,-152,-714,-3472,-17318,-88048,-454378,-2373048,-12515634,-66551016,-356345666,-1919453984,-10392792766,-56527200992,-308691183938,-1691769619240,-9301374102034,-51286672777080,-283527726282794,-1571151822119216,-8725364469143718,-48552769461088336,-270670485377401738,-1511484024051198680,-8453722260102884930,-47350642314439048648,-265579129813183372802,-1491465339550559632448,-8385872784303807639294,-47202746620874986470336,-265975151780412455885826,-1500179080790296495333960,-8469330846027919131108866,-47856040705247407564621400,-270636033194089067428986890};

c[T_]=Sum[cn[[n]]Exp[-2(2(n-1))/T],{n,Length[cn]}];
mag[T_]=Sum[magn[[n]]Exp[-2(2(n-1))/T],{n,Length[cn]}];
chi[T_]=Sum[chin[[n]]Exp[-2(2(n-1))/T],{n,Length[cn]}];

Plot[{c[T],chi[T],mag[T]},{T,0.01,2.4},PlotRange->{-0.3,2},PlotLegends->"Expressions"]

dLogC[T_]=D[Log[c[T]],T];dLogM[T_]=D[Log[mag[T]],T];dLogChi[T_]=D[Log[chi[T]],T];

Tc=2/Log[Sqrt[2]+1]//N (* Exact Tc *)

Plot[{(Tc-T)*dLogChi[T],7/4},{T,0.01,Tc},PlotRange->{-1,5}]

Plot[{-(Tc-T)*dLogM[T],1/8},{T,0.01,Tc},PlotRange->{0,0.13}]

Plot[{(Tc-T)*dLogC[T],0},{T,0.01,Tc},PlotRange->{-1,5}]