Limits

A sequence c1, c2, ... has a limit c if, for any arbitrarily small ϵ>0, there is a subscript n such that for all (Subscript[c, m]) with m⩾n we have |(Subscript[c, m])-c|⩽c.

3/10 + 3/100 + 3/1000;
Sum[3/10^i,{i,1,3}];

f1[i_]:=1/i;
seq1[n_]:=Table[f1[i],{i,1,n}];
ListPlot[seq1[10],AxesOrigin->  {0,0}];
Limit[f1[i],i-> Infinity];

f2[i_]:=Sqrt[i];
seq2[n_]:=Table[f2[i],{i,1,n}];
ListPlot[seq2[100],AxesOrigin->  {0,0}];
Limit[f2[i],i-> Infinity];

f3[i_]:=i/(1+i);
seq3[n_]:=Table[f3[i],{i,1,n}];
ListPlot[seq3[100],AxesOrigin->  {0,0}];
Limit[f3[i],i-> Infinity];

f4[i]:=i/(10+i^2);
seq4[n_]:=Table[f4[i],{i,1,n}];
ListPlot[seq4[10],AxesOrigin->  {0,0}];
Limit[f4[i],i-> Infinity];

f5[i_]:=(i^2+4)/(i^3-10);
seq5[n_]:=Table[f5[i],{i,1,n}];
ListPlot[seq5[100],AxesOrigin->  {0,0}];
Limit[f5[i],i-> Infinity];

f6[i_]:=(10i^2+4)/(i^3+15i^2-10)
seq6[n_]:=Table[f6[i],{i,1,n}];
ListPlot[seq6[50],AxesOrigin->  {0,0}];
Limit[f6[i],i-> Infinity];

f7[i_]:=2^i/Factorial[i];
seq7[n_]:=Table[f7[i],{i,1,n}];
ListPlot[seq7[10],AxesOrigin->  {0,0}];
Limit[f7[i],i-> Infinity];

f8[i_]:=(2^i+i^10)/Factorial[i];
seq8[n_]:=Table[f8[i],{i,1,n}];
ListPlot[seq8[50],AxesOrigin->  {0,0}];
Limit[f8[i],i-> Infinity];

sum1[n_]:=Sum[1/i,{i,1,n}];
sumtab1[n_]:=Table[1/i,{i,1,n}];
ListPlot[sumtab1[20],AxesOrigin->  {0,0}];
sum1[10];

sum2[n_]:=Sum[1/i^2,{i,1,n}];
sumtab2[n_]:=Table[1/i^2,{i,1,n}];
ListPlot[{sumtab1[20],sumtab2[20]},AxesOrigin->  {0,0}];
sum2[10];