An approximation to π

Animate[N[Pi,M], {M,1,2500,1}, AnimationRepetitions -> 1, DefaultDuration -> 250,LabelStyle -> Medium]

Infinite Sums with Alternating Terms

Manipulate[ 
N[ Sum[(-1)^(n+1)/n,{n,1,M}],1000],
{M,1,1000,1, Appearance -> "Labeled"}, LabelStyle -> Medium]

axis :=Plot[.1,{x,0,1}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{.65,.75},{-1.1,.2}}]

int[a_,b_] := Graphics[{Thickness[.015],Line[{{a,.1},{b,.1}}]}]

upper[M_]:=N[Sum[(-1)^(n+1)/n,{n,1,2M-1}]]
lower[M_]:=N[Sum[(-1)^(n+1)/n,{n,1,2M}]]

Slider[Dynamic[M],{1,1000,1}]

Dynamic[Show[axis,int[lower[M],upper[M]]]]

Infinite Sums with Alternating Terms

Manipulate[ 
N[ Sum[(-1)^(n+1)/n!,{n,1,M}],1000],
{M,1,550,1, Appearance -> "Labeled"}, LabelStyle -> Medium]

axis1 :=Plot[.1,{x,0,1}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{.6,.7},{-1.1,.2}}]

upper1[M_]:=N[Sum[(-1)^(n+1)/n!,{n,1,2M-1}]]
lower1[M_]:=N[Sum[(-1)^(n+1)/n!,{n,1,2M}]]

Slider[Dynamic[M1],{1,10,1}]

Dynamic[Show[axis1,int[lower1[M1],upper1[M1]]]]

axis11 :=Plot[.1,{x,0,1}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{.6321,.6322},{-1.1,.2}}]

Dynamic[Show[axis11,int[lower1[M1],upper1[M1]]]]

Mapping intervals

axis21[L_] := Plot[1,{x,0,L}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{0,L},{-1.5,1.5}}]

axis2[L_] := Plot[-1,{x,0,L}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{0,L},{-1.5,1.5}}]

f[x_] :=x^2

Manipulate[
Show[
axis21[L], axis2[L],
Graphics[{Thickness[.015],Line[{{x-ϵ,1},{x+ϵ,1}}]}],
Graphics[{Thickness[.015],Line[{{f[x-ϵ],-1},{f[x+ϵ],-1}}]}],
Graphics[Text["x",{1,1.1}]],
Graphics[Text["f[x]",{1.5,-.9}]]
],
{{x,5,x},0,L, Appearance -> "Labeled"},{{ϵ,1,δ},0,5, Appearance -> "Labeled"},{{L,50},.1,500, Appearance -> "Labeled"},LabelStyle -> Medium]

Uncertainty in Energy

axis31 := Plot[0,{x,0,10}, Axes -> {True,True}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{0,5},{0,5}}]

axis3:= Plot[.1,{x,0,100}, Axes -> {True,False}, PlotStyle -> {RGBColor[0,0,1],Thickness[.01],Opacity[.2]},
PlotRange -> {{0,100},{-1.1,1.2}}]

En[m_,v_] :=(1/2)m v^2

{Slider2D[Dynamic[{m,v}],{1,5}], "(m,v) = "Dynamic[{m,v}]}
{Slider2D[Dynamic[{δ1,δ2}],{.1,1}], "(δ1,δ2) = "Dynamic[{δ1,δ2}]}
Dynamic[Show[axis31,
  Graphics[{
Rotate[Text[Style["Mass", FontSize ->24, FontColor -> Red],{.25,2.5}],Pi/2],
Text[Style["Velocity", FontSize ->24, FontColor -> Red],{2.5,.35}],
Polygon[ {{m-δ1,v-δ2}, {m-δ1, v+δ2},
  {m+δ1,v+δ2}, {m+δ1,v-δ2}}]}]]]
Dynamic[Show[axis3,
Graphics[{Text[Style["Energy", FontSize ->24, FontColor -> Red],{50,-.3}],Thickness[.015],Line[{{En[m-δ1,v-δ2],.1},{En[m+δ1,v+δ2],.1}}]}]]]
Dynamic[{En[m-δ1,v-δ2],En[m+δ1,v+δ2]}]