As doume thn proseggish ths sunarthshs (1/(1 - x))me dunameis gurw apo to x = 0 (order of contact = 8).

Blue = RGBColor[0, 0, 1]; 
    Red = RGBColor[1, 0, 0];
f[x_] = (1/(1 - x));

\(\(approx[x_] = Series[f[x], {x, 0, 8}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-2\), 1}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-5\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=0; Order of Contact=8"];\)\)

H proseggish einai ok sto diasthma [-0.7, 0.7]. As au3hsoume to diasthma me mia proseggish me order of contact 30.

\(\(approx[x_] = Series[f[x], {x, 0, 30}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-2\), 1}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-5\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=0; Order of Contact=30"];\)\)

Twra h proseggish einai ok sto diasthma [-1, 1]. Alla "kolaei" sto -1. Au3anoume thn order of contact sto 100.

\(\(approx[x_] = Series[f[x], {x, 0, 100}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-2\), 1}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-5\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=0; Order of Contact=100"];\)\)

Pali den mporei na perasei to -1. Alla OUTE kai to +1.

\(\(approx[x_] = Series[f[x], {x, 0, 100}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-2\), 5}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-8\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=0; Order of Contact=100"];\)\)

As doume ti ginetai gia anaptugmata gurw apo to x=5:

\(\(approx[x_] = Series[f[x], {x, 5, 100}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-2\), 15}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-8\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=5; Order of Contact=100"];\)\)

kai gurw apo to x=-5. (Order of Contact 100 kai stis 2 periptwseis!) :

\(\(approx[x_] = Series[f[x], {x, \(-5\), 100}] // Normal;\)
\)
\(\(Plot[{f[x], approx[x]}, {x, \(-15\), 3}, 
	
        PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.01], Red}}, 
        AspectRatio -> 1\/GoldenRatio, PlotRange -> {\(-8\), 8}, 
        AxesLabel -> {"x", ""}, 
        PlotLabel -> "Powers of x centered at x=-5; Order of Contact=100"];\)\)

Blepoume pws otan anaptusoume gurw apo to shmeio x=b, tote to b einai sto kentro 2 fragmatwn!  Ena fragma einai panta sto x=1 kai to allo fragma einai sto shmeio x = 2  b - 1, toso gia b>0 oso kai gia b<0.