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EXAMPLE 18 To obtain the Maclaurin polynomial of degree 5 for the function f ( x ) = e x , we can use the
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{x,k}f[x] /. x 0 k x k! k=0
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Mathematica includes a convenient command for constructing the Taylor polynomial.
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Series[f[x], {x, a, n}] generates a SeriesData object2 representing the nth degree Taylor polynomial of f(x) about a.
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f[x_]= Exp[x]; Series[f[x], {x, 0, 5}] 2 3 4 5 1 + x + x + x + x + x + O[x]6 2 6 24 120 The symbol O[x]6 in the above expansion represents the order of the omitted terms in the (infinite) expansion. O[x]6 means that the omitted terms have powers of x of degree 6.
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We can see what a SeriesData object looks like by using the command InputForm.
InputForm[expression] prints expression in a form suitable for input to Mathematica.
1 An analytic function of a real variable is one that has a Taylor series expansion. Most functions encountered in applications are of this type; however, even if a function has derivatives of all orders, it may not be analytic. 2 A SeriesData object is a representation of a power series but does not have a numerical value.
Differential Calculus
EXAMPLE 20
f[x_]= Exp[x]; s = Series[f[x], {x, 0, 5}]; InputForm[s] SeriesData[x, 0, {1, 1, 1/2, 1/6, 1/24, 1/120}, 0, 6, 1]
A SeriesData object is non-numerical and therefore cannot be evaluated numerically.
EXAMPLE 21
f[x_] = Exp[x]; p[x_] = Series[f[x], {x, 0, 5}]
2 3 4 5 1 + x + x + x + x + x + O[x]6 2 6 24 120 p[1]
SeriesData ssdn : Attempt to evaluate a series at the number 1. Returning Indeterminate. Indeterminate
In order to convert the series into one that can be evaluated, the function Normal can be used to transform it into an ordinary polynomial.
Normal[series] returns a polynomial representation of the SeriesData object series, which can then be evaluated numerically. The O[x]n term is omitted.
EXAMPLE 22
f[x_] = Exp[x]; s = Series[f[x], {x, 0, 5}]
2 3 4 5 1 + x + x + x + x + x + O[x]6 2 6 24 120
p[x_] = Normal[s]
2 3 4 5 1+ x + x + x + x + x 2 6 24 120
Normal has converted the SeriesData object into an ordinary polynomial, whose value can now be computed.
p[1] 163 60
The number obtained in the previous example, 163/60, is approximately 2.71667. If we compare this to the (known) value of e 2.71828, we see a small error in our approximation. We would expect the error to diminish as the degree of the polynomial increases. This is shown to be the case in the next example.
EXAMPLE 23
f[x_] = Exp[x] ; exactvalue = f[1] ; p[n_] Normal[Series[f[x], {x, 0, n}]] /. x 1 data = Table[{n, N[p[n]], N[Abs[p[n] exactvalue]]}, {n, 1, 10}];
Differential Calculus
TableForm[data, TableSpacing {1,10}, TableHeadings {None, {"n", "
n 1 2 3 4 5 6 7 8 9 10 p(1) 2. 2.5 2.66667 2.70833 2.71667 2.71806 2.71825 2.71828 2.71828 2.71828 Error 0.718282 0.218282 0.0516152 0.0099485 0.00161516 0.000226273 0.0000278602 3.05862 10 6 3.02886 10 7 2.73127 10 8
p(1)","
Error "}}]
EXAMPLE 24 To see the convergence of a power series even more dramatically, we can construct an animation showing the sequence of Maclaurin polynomials converging to ex. We consider the interval [0, 5].
f[x_] = Exp[x]; p[n_, x_] Normal[Series[f[t], {t, 0, n}]] /.t x Animate[Plot[{p[n, x], f[x]}, {x, 0, 5}, PlotRange {0,Exp[5]}], {n, 1, 10, 1}]
Output of Animate when n = 5
If only the coefficient of a particular term of a series is needed, the command SeriesCoefficient may be used. The actual series, which may be quite long, need not be printed in its entirety. SeriesCoefficient is the SeriesData equivalent of Coefficient for polynomials.
SeriesCoefficient[series, n] returns the coefficient of the nth degree term of a SeriesData object.
EXAMPLE 25
f[x_] = Exp[x] ; s = Series[f[x], {x, 0, 10}]; SeriesCoef cient[s, 10] 1 3 628 800
Differential Calculus
SOLVED PROBLEMS
8.20 Obtain the Maclaurin polynomial of degree 10 for the function f(x) = tan 1x by using a direct summation and then by using the Series command.
SOLUTION
f[x_] = ArcTan[x];
Derivative[k][f][0]x k!
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