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Compute Factorial Power Series Approximations
Derive approximations for functions using factorial power series.
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FactorialSeries[f_, {x_, x0_, n_, h_ : 1}] := \!\(

\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n\)]\(

FractionBox[\(\((

\*SubscriptBox[\(\[DifferenceDelta]\), \({x, i, h}\)]f /.  

      x -> x0)\)\(\ \)\), \(

\*SuperscriptBox[\(h\), \(i\)]\ \(i!\)\)] FactorialPower[x - x0, i, 

    h]\)\)
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FactorialSeries[Sin[x/5], {x, 0, 3}] // TraditionalForm
Out[2]//TraditionalForm=
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Plot[Evaluate[

  Table[FactorialSeries[Sin[x/5], {x, 0, n}], {n, 1, 

    7}]], {x, -5 \[Pi], 5 \[Pi]}, 

 Ticks -> {{-5 \[Pi], 0, 5 \[Pi]}, Automatic}]
Out[3]=