The function f:(0,∞)⟶ℝ, f(x)=(1/(1-x)) has Taylor Series f(x)=(∑_(i=0)^∞)(((-1))^i)(((x-2))^i)at x = 2.  It has a radius of convergence of 1.  Here we plot f(x) along with a few of its Taylor polynomials.

T0=Plot[-1,{x,0,5},PlotStyle->{Green,Thick}];
T1=Plot[-1+(x-2),{x,0,5},PlotStyle->{Red,Thick}];
T2=Plot[-1+(x-2)-(x-2)^2,{x,0,5},PlotStyle->{Orange,Thick}];
T3=Plot[-1+(x-2)-(x-2)^2+(x-2)^3,{x,0,5},PlotStyle->{Black,Thick}];
T4=Plot[-1+(x-2)-(x-2)^2+(x-2)^3-(x-2)^4,{x,0,5},PlotStyle->{Green,Thick}];
T5=Plot[-1+(x-2)-(x-2)^2+(x-2)^3-(x-2)^4+(x-2)^5,{x,0,5},PlotStyle->{Red,Thick}];
T6=Plot[-1+(x-2)-(x-2)^2+(x-2)^3-(x-2)^4+(x-2)^5-(x-2)^6,{x,0,5},PlotStyle->{Orange,Thick}];

T15=Plot[-1+(x-2)-(x-2)^2+(x-2)^3-(x-2)^4+(x-2)^5-(x-2)^6+(x-2)^7-(x-2)^8+(x-2)^9-(x-2)^10+(x-2)^11-(x-2)^12+(x-2)^9-(x-2)^13+(x-2)^14-(x-2)^15,{x,0,5},PlotStyle->{Orange,Thick}];
T32=Plot[-1+(x-2)-(x-2)^2+(x-2)^3-(x-2)^4+(x-2)^5-(x-2)^6+(x-2)^7-(x-2)^8+(x-2)^9-(x-2)^10+(x-2)^11-(x-2)^12+(x-2)^13-(x-2)^14+(x-2)^15-(x-2)^16+(x-2)^17-(x-2)^18+(x-2)^19-(x-2)^20+(x-2)^21-(x-2)^22+(x-2)^23-(x-2)^24+(x-2)^25-(x-2)^26+(x-2)^27-(x-2)^28+(x-2)^29-(x-2)^30+(x-2)^31-(x-2)^32,{x,.9,3.1},PlotStyle->{Orange,Thick}];

FncPlot=Show[Plot[1/(1-x),{x,1,5},PlotRange->{{0,5},{-8,2}},PlotStyle->Thickness[.005]],
ParametricPlot[{1,t},{t,-8,2}],ParametricPlot[{3,t},{t,-8,2}],Graphics[{Pink,Opacity[0.5],Rectangle[{0,-8},{1,2}]}],Graphics[{Pink,Opacity[0.5],Rectangle[{3,-8},{5,2}]}],ImageSize->Scaled[.55]]

Print[Style["0th",14,Bold,Green]," degree Taylor Polynomial: ",Show[FncPlot,T0]]
Print[Style["0th",14,Bold,Green],", ",Style["1st",14,Bold,Red]," degree Taylor Polynomials: ",Show[FncPlot,T0,T1]]
Print[Style["0th",14,Bold,Green],", ",Style["1st",14,Bold,Red],", ",Style["2nd",14,Bold,Orange]," degree Taylor Polynomials: ",Show[FncPlot,T0,T1,T2]]
Print[Style["0th",14,Bold,Green],", ",Style["1st",14,Bold,Red],", ",Style["2nd",14,Bold,Orange],", ",Style["3rd",14,Bold,Black]," degree Taylor Polynomials: ",Show[FncPlot,T0,T1,T2,T3]]
Print[Style["4th",14,Bold,Green]," degree Taylor Polynomial: ",Show[FncPlot,T4]]
Print[Style["4th",14,Bold,Green],", ",Style["5th",14,Bold,Red]," degree Taylor Polynomials: ",Show[FncPlot,T4,T5]]
Print[Style["4th",14,Bold,Green],", ",Style["5th",14,Bold,Red],", ",Style["6th",14,Bold,Orange]," degree Taylor Polynomials: ",Show[FncPlot,T4,T5,T6]]
Print[Style["15th degree Taylor Polynomial: ",14,Bold,Orange],Show[FncPlot,T15]]
Print[Style["31st degree Taylor Polynomial: ",14,Bold,Orange],Show[FncPlot,T32]]

The function i:ℝ⟶ℝ, i(x)=sin x has Taylor Series i(x)=(∑_(i=0)^∞)(((-1))^i)((x^(2i+1))/((2i+1)!))at x = 0.  It has an infinite radius of convergence.  Here again we plot i(x) along with a few of its Taylor polynomials.

ix=Plot[Sin[x],{x,-10,10},PlotRange->{{-10,10},{-3,3}},PlotStyle->{Thickness[.006],Red},ImageSize->Scaled[.6],Frame->True]

s1=Plot[(x)/1 ,{x,-10,10},PlotStyle->{Thick,Blue}];
s3=Plot[(x)/1 - (x^3)/3!,{x,-10,10},PlotStyle->{Thick,Green}];
s5=Plot[(x)/1 - (x^3)/3! + (x^5)/5!,{x,-10,10},PlotStyle->{Thick,Orange}];
s7=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7!,{x,-10,10},PlotStyle->{Thick,Blue}];
s9=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7!+ (x^9)/9!,{x,-10,10},PlotStyle->{Thick,Green}];
s11=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9!- (x^11)/11!,{x,-10,10},PlotStyle->{Thick,Orange}];
s13=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! ,{x,-10,10},PlotStyle->{Thick,Blue}];
s15=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! - (x^15)/15!,{x,-10,10},PlotStyle->{Thick,Green}];
s17=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! - (x^15)/15! + (x^17)/17!,{x,-10,10},PlotStyle->{Thick,Orange}];
s19=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! - (x^15)/15! + (x^17)/17!- (x^19)/19! ,{x,-10,10},PlotStyle->{Thick,Blue}];
s21=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! - (x^15)/15! + (x^17)/17!- (x^19)/19! + (x^21)/21!  ,{x,-10,10},PlotStyle->{Thick,Green}];
s23=Plot[(x)/1 - (x^3)/3! + (x^5)/5!- (x^7)/7! + (x^9)/9! - (x^11)/11! + (x^13)/13! - (x^15)/15! + (x^17)/17!- (x^19)/19! + (x^21)/21! -(x^23)/23! ,{x,-10,10},PlotStyle->{Thick,Orange}];

Print[Style["1st",14,Bold,Blue],", ",Style["3rd",14,Bold,Green],", ",Style["5th",14,Bold,Orange]," degree Taylor Polynomials: ",Show[ix,s1,s3,s5]]
Print[Style["7th",14,Bold,Blue],", ",Style["9th",14,Bold,Green],", ",Style["11th",14,Bold,Orange]," degree Taylor Polynomials: ",Show[ix,s7,s9,s11]]
Print[Style["13th",14,Bold,Blue],", ",Style["15th",14,Bold,Green],", ",Style["17th",14,Bold,Orange]," degree Taylor Polynomials: ",Show[ix,s13,s15,s17]]
Print[Style["19th",14,Bold,Blue],", ",Style["21st",14,Bold,Green],", ",Style["23rd",14,Bold,Orange]," degree Taylor Polynomials: ",Show[ix,s19,s21,s23]]