nmax=100;
f[x_]:=1/(1-x);
Sf[x_,n_]:=Sum[x^k,{k,0,n}];
(*Do[Print["n=",n];
Plot[{f[x],Sf[x]},{x,-2,2},PlotRange->{-10,10}]
,{n,0,nmax}];*)
Manipulate[plotlabel=Column[{f[x],ToString[Sum[x^k,{k,0,n}]]}];
Plot[{f[x],Sf[x,n]},{x,-2,2},PlotRange->{-10,10},PlotLabel->plotlabel]
,{n,0,nmax,1}]

nmax=100;
f[x_]:=1/(1+x);
Sf[x_,n_]:=Sum[(-x)^k,{k,0,n}];
(*Do[Print["n=",n];
Plot[{f[x],Sf[x]},{x,-2,2},PlotRange->{-10,10}]
,{n,0,nmax}];*)
Manipulate[plotlabel=Column[{f[x],plotlabel=Sf[x,n]}];
Plot[{f[x],Sf[x,n]},{x,-2,2},PlotRange->{-10,10},PlotLabel->plotlabel]
,{n,0,nmax,1}]

nmax=100;
a[n_]:=(1/(2n-1))*(-1)^(n+1);
Table[a[n],{n,1,5}]
f[x_]:=ArcTan[x];
Sf[x_,n_]:=Sum[a[k]*(x)^(2k-1),{k,1,n}];
(*Do[Print["n=",n];
Plot[{f[x],Sf[x]},{x,-2,2},PlotRange->{-10,10}]
,{n,0,nmax}];*)
Manipulate[plotlabel=Column[{f[x],plotlabel=Sf[x,n]}];
Plot[{f[x],Sf[x,n]},{x,-1.5,1.5},PlotRange->{{-1.5,1.5},{-Pi/2,Pi/2}},PlotLabel->plotlabel]
,{n,1,nmax,1}]

nmax=100;
a[n_]:=(-1/n)*(-1)^n;
Table[a[n],{n,1,5}]
f[x_]:=Log[x+1];
Sf[x_,n_]:=Sum[a[k]*(x^k),{k,1,n}];
(*Do[Print["n=",n];
Plot[{f[x],Sf[x]},{x,-2,2},PlotRange->{-10,10}]
,{n,0,nmax}];*)
Manipulate[plotlabel=Column[{f[x],plotlabel=Sf[x,n]}];
Plot[{f[x],Sf[x,n]},{x,-10,10},PlotRange->{{-2,4},{-10,10}},PlotLabel->plotlabel]
,{n,1,nmax,1}]

nmax=100;
a[n_]:=(1/n!);
Table[a[n],{n,1,5}]
f[x_]:=Exp[x];
Sf[x_,n_]:=Sum[a[k]*(x^k),{k,0,n}];
(*Do[Print["n=",n];
Plot[{f[x],Sf[x]},{x,-2,2},PlotRange->{-10,10}]
,{n,0,nmax}];*)
Manipulate[plotlabel=Column[{f[x],plotlabel=Sf[x,n]}];
Plot[{f[x],Sf[x,n]},{x,-10,10},PlotRange->{{-2,4},{-10,10}},PlotLabel->plotlabel]
,{n,1,nmax,1}]

nmax=50;
Tlog[x_,n_]:=Sum[(1/k) (-1)^(k-1) (x-1)^k,{k,1,n}];
Manipulate[plotlabel=Column[{Log[x],plotlabel=Tlog[x,n]}];
(*"n="<>ToString[n]*);
Plot[{Tlog[x,n],Log[x]},{x,0.001,3},PlotLabel->plotlabel,
PlotRange->{-7.5,7.5}],{n,1,nmax,1}
]