SetOptions[ListPlot,PlotMarkers->{Automatic,Medium}];

Plot[{1/(1-x), Sum[x^n,{n,0,10}]},{x,-2,2},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,20},
Epilog->{
Thick,Dotted,
InfiniteLine[{-1,0},{0,1}],
InfiniteLine[{1,0},{0,1}]
}
]

Plot[{1/(x-3), Sum[(-1)/(3^(n+1)) x^n,{n,0,1000}]},{x,-4,4},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,10},
Epilog->{
Thick,Dotted,
InfiniteLine[{-3,0},{0,1}],
InfiniteLine[{3,0},{0,1}]
}
]

Plot[{1/(x-3), Sum[(-1)^n (x-4)^n,{n,0,1000}]},{x,0,6},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,10},
Epilog->{
Thick,Dotted,
InfiniteLine[{3,0},{0,1}],
InfiniteLine[{5,0},{0,1}]
}
]

Plot[{Log[1+x], Sum[(-1)^(n-1)x^n/n,{n,1,10}]},{x,-2,2},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,5},
Epilog->{
Thick,Dotted,
InfiniteLine[{-1,0},{0,1}],
InfiniteLine[{1,0},{0,1}]
}
]

Plot[{ArcTan[x], Sum[(-1)^(n)x^(2n+1)/(2n+1),{n,0,500}]},{x,-2,2},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-5,5},
Epilog->{
Thick,Dotted,
InfiniteLine[{-1,0},{0,1}],
InfiniteLine[{1,0},{0,1}]
}
]

Plot[{Log[x], Sum[(-1)^(n-1)(x-1)^(n)/(n),{n,1,10}]},{x,-1,3},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,5},
Epilog->{
Thick,Dotted,
InfiniteLine[{0,0},{0,1}],
InfiniteLine[{2,0},{0,1}]
}
]

Plot[{Log[x], Log[5]+Sum[(-1)^(n-1)(x-5)^(n)/(n 5^n),{n,1,10}]},{x,-2,12},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-10,5},
Epilog->{
Thick,Dotted,
InfiniteLine[{0,0},{0,1}],
InfiniteLine[{10,0},{0,1}]
}
]

Plot[{Sin[x], Sum[(-1)^(n)(x)^(2n+1)/((2n+1)!),{n,0,11}]},{x,-4Pi,4Pi},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-2,2},
Epilog->{
Thick,Dotted,
InfiniteLine[{2Pi,0},{0,1}],
InfiniteLine[{-2Pi,0},{0,1}],
InfiniteLine[{4Pi,0},{0,1}],
InfiniteLine[{-4Pi,0},{0,1}]
}
]

Plot[{Cos[x], Sum[(-1)^(n)(x)^(2n)/((2n)!),{n,0,12}]},{x,-4Pi,4Pi},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{-2,2},
Epilog->{
Thick,Dotted,
InfiniteLine[{2Pi,0},{0,1}],
InfiniteLine[{-2Pi,0},{0,1}],
InfiniteLine[{4Pi,0},{0,1}],
InfiniteLine[{-4Pi,0},{0,1}]
}
]

Plot[{(1+x^2)^(1/3), Sum[Binomial[1/3,n]x^(2n),{n,0,1}]},{x,-2,2},PlotStyle->{Thick,Directive[Dashed,Thick,Red]},PlotRange->{0,2},
Epilog->{
Thick,Dotted,
InfiniteLine[{-1,0},{0,1}],
InfiniteLine[{1,0},{0,1}]
}
]

f[x_]:=E^(-x^2/2)/Sqrt[2 Pi]
Plot[{f[x],Piecewise[{{f[x],-1<=x<=1}},_]},{x,-3,3},
Filling->{2->Axis}]
Plot[{E^(-x^2),Piecewise[{{E^(-x^2),0<=x<=1}},_]},{x,-3,3},
Filling->{2->Axis}]

f[x_]:=Sin[2x]-x
Solve[f'[x]==0,x]
a0=-.3
a1=Pi/6
a2=-Pi/6
Plot[{
f[x],
f[a1]+f'[a1](x-a1),
f[a0]+f'[a0](x-a0)
},
{x,-Pi/2,Pi/2},PlotStyle->Thick]
Plot[{
f[x],
f[a1]+f'[a1](x-a1) + f''[a1]/2 (x-a1)^2,
f[a2]+f'[a2](x-a2)+f''[a2]/2(x-a2)^2
},
{x,-Pi/2,Pi/2},PlotStyle->Thick]

f[a1]+f'[a1](x-a1) + f''[a1]/2 (x-a1)^2

f[x_]:=Cos[x^2]
Solve[f'[x]==0,x]
a0=0
a1=Pi/6
a2=-Pi/6
Plot[{
f[x],
f[a0]+f'[a0](x-a0)+f''[a0]/2(x-a0)^2
},
{x,-Pi/2,Pi/2},PlotStyle->Thick]
Plot[{
f[x],
f[a0]+f'[a0](x-a0)+f''[a0]/2(x-a0)^2 + f'''[a0]/6(x-a0)^3 + f''''[a0]/24(x-a0)^4
},
{x,-Pi/2,Pi/2},PlotStyle->Thick]

f[x_]:=Sin[x]-x
Solve[f'[x]==0,x]
a0=0
a1=Pi/6
a2=-Pi/6
Plot[{
f[x],
f[a0]+f'[a0](x-a0)+f''[a0]/2(x-a0)^2
},
{x,-Pi/2,Pi/2},PlotStyle->Thick,PlotRange->{-3/4,3/4}]
Plot[{
f[x],
f[a0]+f'[a0](x-a0)+f''[a0]/2(x-a0)^2 + f'''[a0]/6(x-a0)^3 
},
{x,-Pi/2,Pi/2},PlotStyle->Thick,PlotRange->{-3/4,3/4}]

Plot[{E^(-1/x^2)},{x,-1,1},PlotStyle->Thick]

https://www.youtube.com/watch?v=ZY6h3pKqYI0
(2:50 , 4:32)

https://staff.tumblr.com/post/135408432752/red3blog-formeldeharv-i-put-all-i-want-for