In[]:=
l2=Table[Re[NSum[(-1)^x(-1),{x,1,Infinity,1/2^n},WorkingPrecision->50,Method->"AlternatingSigns"]],{n,30,60}];
1/x
x
In[]:=
l2a=Ratios[l2]
Out[]=
{0.500000002323121059881671913315479168849495407160,0.500000001161560489674947973048611728299794218010,0.500000000580780234771002004607967566476060025251,0.500000000290390114868883008573141918954119507157,0.500000000145195056805287006072390816662892819787,0.500000000072597528245354878509966662065630518968,0.500000000036298764083355283126840680585491048686,0.500000000018149382031847102531811494692019068861,0.500000000009074691013465916508056888071979331273,0.500000000004537345506118549564572898233236249690,0.500000000002268672752905672609923396292898113245,0.500000000001134336376414435761871039143890038742,0.500000000000567168188197617745162867846726634417,0.500000000000283584094096408838638272620236391934,0.500000000000141792047047604410833346187858673505,0.500000000000070896023523652203295225588804732243,0.500000000000035448011761788601117250921301249507,0.500000000000017724005880884925426034992772849914,0.500000000000008862002940440118929869879466669287,0.500000000000004431001470219473519148035509606731,0.500000000000002215500735109590273127291699647763,0.500000000000001107750367554758514951964336132028,0.500000000000000553875183777370102073061789655181,0.500000000000000276937591888682762185800800226399,0.500000000000000138468795944340808880217876463091,0.500000000000000069234397972170261386938307319042,0.500000000000000034617198986085094930176495931398,0.500000000000000017308599493042538524265083533669,0.500000000000000008654299746521267026926750658827,0.500000000000000004327149873260632954661927552412}
In[]:=
l2b=Ratios[l2a-1/2]
Out[]=
{0.499999982667331169706640515776651602643,0.499999991333665296472551369843345133785,0.499999995666832576141079391164782565651,0.49999999783341627004674011317493119838,0.49999999891670813051742009736456888132,0.49999999945835406413222255092393117418,0.4999999997291770317844894000282986268,0.4999999998645885158218392310314727424,0.4999999999322942578933182482545347495,0.499999999966147128942258782310025409,0.499999999983073564470029305700459518,0.499999999991536782234739631486561126,0.499999999995768391117301060402359612,0.49999999999788419555863334136594909,0.49999999999894209777931237347416681,0.49999999999947104888965511243488146,0.4999999999997355244448272876418902,0.4999999999998677622224135766770575,0.4999999999999338811112067715525568,0.499999999999966940555603381579785,0.499999999999983470277801689740769,0.499999999999991735138900844608104,0.499999999999995867569450422238482,0.49999999999999793378472521110285,0.49999999999999896689236260554733,0.49999999999999948344618130277264,0.4999999999999997417230906513861,0.4999999999999998708615453256930,0.4999999999999999354307726628465}
In[]:=
Ratios[l2b-1/2]
Out[]=
{0.5000000166379898970909991437720,0.500000008318995141557602429649,0.500000004159497619031827992458,0.50000000207974882157917082354,0.50000000103987441380539963521,0.5000000005199372076566533755,0.500000000259968604016815078,0.500000000129984302055529636,0.50000000006499215103954534,0.50000000003249607552271780,0.5000000000162480377620952,0.500000000008124018881232,0.500000000004062009440662,0.50000000000203100472034,0.50000000000101550236017,0.5000000000005077511801,0.500000000000253875590,0.500000000000126937795,0.50000000000006346890,0.50000000000003173445,0.5000000000000158672,0.500000000000007934,0.500000000000003967,0.50000000000000198,0.50000000000000099,0.5000000000000005,0.500000000000000,0.500000000000000}
In[]:=
Table[Re[NSum[(-1)^x(-1),{x,1,Infinity,1/5^n},WorkingPrecision->50,Method->"AlternatingSigns"]],{n,10,30}]
1/x
x
Out[]=
{-0.00001149438825589711630737193829019531497942306889296,-2.298879530237538529998025832022468379653615147007×,-4.59775981209514772239246687294395345764602037277×,-9.19551992483806967379127934674137780935886107625×,-1.839103996993522871630330045968981599967518737068×,-3.67820799879740915542642640065663934104796751058×,-7.35641599951896366271251411200419708025234833478×,-1.471283199980758546512836112895956242124257046582×,-2.94256639999230341860548128903167526199662581513×,-5.88513279999692136744222026312253400111969588259×,-1.177026559999876854697689032504980271968801283032×,-2.354053119999950741879075630760398421740037143426×,-4.70810623999998029675163025372483187370527458278×,-9.41621247999999211870065210160358654982858127771×,-1.883242495999999684748026084065052692385060538181×,-3.76648499199999987389921043362609381536366861939×,-7.53296998399999994955968417345044334517482104222×,-1.506593996799999997982387366938017780350544078872×,-3.01318799359999999919295494677520711584633454738×,-6.02637598719999999967718197871008284660028880593×,-1.205275197439999999987087279148403313859158718874×}
-6
10
-7
10
-8
10
-8
10
-9
10
-10
10
-10
10
-11
10
-12
10
-12
10
-13
10
-14
10
-15
10
-15
10
-16
10
-17
10
-17
10
-18
10
-19
10
-19
10
Consider this list.
In[]:=
N[Table[Re[NSum[(-1)^x(-1),{x,1,Infinity,1/10^n},WorkingPrecision->100,Method->"AlternatingSigns"]],{n,1,20}],50]
1/x
x
Out[]=
{1.9188586179657711644749569365028543108104022761266,-0.96896118100337209404725066070509797376399062869078,-0.11108252178329225990690716783464721040675606141553,-0.011213751931258626786940085449600662574200338892350,-0.0011223879518722938322756578600167206664824970434668,-0.00011224887995186639828880804717545957545481269743429,-0.000011224988799951865808743881014870954235434222441793,-1.1224998879999518657497894825688853901707392782723×,-1.1224999887999995186574389404366691924624797685684×,-1.1224999988799999951865743304499786149015944793966×,-1.1224999998879999999518657432455453981662568153309×,-1.1224999999887999999995186574323964995936892354145×,-1.1224999999988799999999951865743239060415489199640×,-1.1224999999998879999999999518657432390014611012273×,-1.1224999999999887999999999995186574323899556566243×,-1.1224999999999988799999999999951865743238994976119×,-1.1224999999999998879999999999999518657432389949172×,-1.1224999999999999887999999999999995186574323899491×,-1.1224999999999999988799999999999999951865743238995×,-1.1224999999999999998879999999999999999518657432390×}
-6
10
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
-13
10
-14
10
-15
10
-16
10
-17
10
-18
10
Look at all the repeating numbers the error of the following approximation to the last element of that list gives.
In[]:=
a=Re[NSum[(-1)^x(-1),{x,1,Infinity,1/10^20},WorkingPrecision->60,Method->"AlternatingSigns"]]
1/x
x
Out[]=
-1.1224999999999999998879999999999999999518657432404630328525×
-18
10
In[]:=
b=N,60
201601
17960000000000000017920
Out[]=
1.12249999999999999888000000000000000111750556792873051113443×
-17
10
In[]:=
NumberForm[-a+b,60,ScientificNotationThreshold->{-40,8}]
Out[]//NumberForm=
0.0000000000000000123474999999999999886880000000000000111269214225277681441967
Also, their sum has all the repeating numbers of their difference.
In[]:=
NumberForm[a+b,60,ScientificNotationThreshold{-40,8}]
Out[]//NumberForm=
0.0000000000000000101024999999999999889120000000000000112231899360468420784918