Series
ComposeSeries

Calculus
Mathematics

Explore Taylor Series of Composed Functions

Compute and visualize a Taylor series approximation to a composite function

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Define transcendental functions , ℊ, and  as below. Notice that

(x)=(ℊ(x))

In[1]:=

=xExp[x];

In[2]:=

ℊ=x-Sin[x^2];

In[3]:=

=x(@*ℊ)[x];

Show , ℊ, and  in traditional notation:

In[4]:=

Comap[{,ℊ,},x]

Out[4]=





,-Sin[

-Sin[

]





Compute Taylor series approximations of  and ℊ:

In[5]:=

seriesF=Series[[x],{x,0,6}]

Out[5]=

1+x+

120

720

O[x]

In[6]:=

seriesG=Series[ℊ[x],{x,0,6}]

Out[6]=

O[x]

Compose the two series, keeping terms to the highest order specified:

In[7]:=

ComposeSeries[seriesF,seriesG]

Out[7]=

O[x]

Verify the previous result by directly computing the Taylor series for  to the same order:

In[8]:=

Series[[x],{x,0,6}]

Out[8]=

O[x]

Define an expression suitable for plotting the

-th order approximation to the function :

In[9]:=

approx[n_]:=Normal@Series[[x],{x,0,n}]

Visualize the fact that higher order series provide better approximations to the function within the series' radius of convergence:

In[10]:=

Plot[Evaluate@{[x],approx[2],approx[4]},{x,-5,5},PlotRange{0,},PlotLegends"Expressions"]

Out[10]=

Related Symbols

Series
ComposeSeries

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Contributed by: John McNally

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Explore Taylor Series of Composed Functions

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