Taylor Series

In Mathematica it is quite easy to capture and plot Taylor polynomials:

Series[Tan[x], {x,0,7}]

To remove the big Oh term use the Normal command:

Normal[%]

To plot a Taylor series not centered at the origin:

G=Normal[Series[Log[x], {x,2,3}]]

Plot[{Log[x],G}, {x, 0,7}]

gaussFunction= ((Exp[-((x^2)/2)])/(Sqrt[2 Pi]))

taylor2=Normal[Series[gaussFunction, {x,1,2}]]

taylor3=Normal[Series[gaussFunction, {x,1,3}]]

taylor4=Normal[Series[gaussFunction, {x,1,4}]]

Plot[{gaussFunction, taylor4, taylor3, taylor2}, {x,-2,3}]

Next, let's  explore  Fourier polynomials that generate a square wave

wave2=Sum[((Sin[(2 n+1)x])/(2n+1)), {n,0,2}]

Plot[wave2,{x,-7,7}]

wave4=Sum[((Sin[(2 n+1)x])/(2n+1)), {n,0,4}]

Plot[wave4,{x,-7,7}]

wave8=Sum[((Sin[(2 n+1)x])/(2n+1)), {n,0,8}]

Plot[wave8,{x,-7,7}]

wave25=Sum[((Sin[(2 n+1)x])/(2n+1)), {n,0,25}]

Plot[wave25,{x,-7,7}]

Products

finite products

Product[i^2,{i,1,6}]

Product[1-(1.0/(j!)),{j,2,11}]

infinite products

Product[1.0-(1/(2^j)),{j,2,∞}]

Product[1-(1.0/(Exp[j])),{j,2,∞}]

ListPlot[Table[Product[1-(1.0/(2^j)),{j,2,n}], {n, 1, 35}], PlotStyle->{Blue, PointSize[0.02]},  AxesOrigin->{0,0.57}]