Taylor Series: Mathematical Background
Definitions
Let be a smooth (differentiable) function, and let , then a Taylor series of the function around the point is given by:
In particular, if , then the expansion is known as the Maclaurin series and thus is given by:
Taylor’s Theorem
Many of the numerical analysis methods rely on Taylor’s theorem. In this section, a few mathematical facts are presented which serve as the basis for Taylor’s theorem. The ideas within the proofs presented here are attributed to Paul’s online calculus notes.
Extreme Values of Smooth Functions
Definition: Local Maximum and Local Minimum
Let . is said to have a local maximum at a point if there exists an open interval such that and . On the other hand, is said to have a local minimum at a point if there exists an open interval such that and . If has either a local maximum or a local minimum at , then is said to have a local extremum at .
Proposition 1
Let be smooth (differentiable). Assume that has a local extremum (maximum or minimum) at a point , then . This proposition is also referred to in some texts as Fermat’s theorem.
View Proof of Proposition 1
If we restrict ourselves to positive then we have:
By definition, the limit from the right is
If we now restrict ourselves to negative then we have:
By definition, the limit from the left is
The basic assumption for the theorem is that the limit at exists. And since that implies the limit from the left is equal to the limit from the right, therefore:
Similar arguments apply if we assume to be a local minimum.
This proposition simply means that if a smooth function attains a local maximum or minimum at a particular point, then the slope of the function is equal to zero at this point.
As an example, consider the function with the relationship . In this case, is a local maximum value for attained at and is a local minimum value of attained at . These local extrema values are associated with a zero slope for the function since
and are locations of local extrema and for both we have . The red lines in the next figure show the slope of the function at the extremum values.
View Mathematica Code that Generated the Above FigureClear[x] y = x^3  3 x; Plot[y, {x, 3, 3}, Epilog > {PointSize[0.04], Point[{1, 2}], Point[{1, 2}], Red, Line[{{3, 2}, {1.5, 2}}], Line[{{3, 2}, {1.5, 2}}]}, Filling > Axis, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}]
import numpy as np import matplotlib.pyplot as plt x = np.arange(3,3,0.01) y = x**3  3*x plt.plot(x,y) plt.fill_between(x, y, 0, alpha=0.20) plt.plot([3,1.5],[2,2],'r') plt.plot([3,1.5],[2,2],'r') plt.plot([1,1],[2,2],'ko') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show()
“Smoothness” or “Differentiability” is a very important requirement for the proposition to work. As an example, consider the function defined as . The function has a local minimum at , however, is not defined as the slope as from the right is different from the slope as from the left as shown in the next figure.
Clear[x] y = Abs[x]; Plot[y, {x, 1, 1}, Epilog > {PointSize[0.04], Point[{0, 0}]}, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}]
import numpy as np import matplotlib.pyplot as plt x = np.arange(1,1,0.01) y = abs(x) plt.plot(x,y) plt.plot([0],[0],'ko') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show()
Extreme Value Theorem
Statement: Let be continuous. Then, attains its maximum and its minimum value at some points and in the interval .
The theorem simply states that if we have a continuous function on a closed interval , then the image of contains a maximum value and a minimum value within the interval . The theorem is very intuitive. However, the proof is highly technical and relies on fundamental concepts in Real analysis including the definitions of real numbers and on continuous functions. You can review the Wikipedia entry or a course on Real analysis such as this one for details of the proof. For now, we will just illustrate the meaning of the theorem using an example. Consider the function defined as:
The theorem states that has to attain a maximum value and a minimum value at a point within the interval. In this case, is the maximum value of attained at and is the minimum value of attained at . Alternatively, if with the same relationship as above, is the minimum value of attained at and is the maximum value of attained at
The following figure shows the graph of the function on the specified intervals.
Clear[x] y = x^3  3 x; Plot[y, {x, 1.5, 1.5}, Epilog > {PointSize[0.04], Point[{1, 2}], Point[{1, 2}]}, Filling > Axis, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}] Plot[y, {x, 3, 3}, Epilog > {PointSize[0.04], Point[{3, y /. x > 3}], Point[{3, y /. x > 3}]}, Filling > Axis, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}]
import numpy as np import matplotlib.pyplot as plt x1 = np.arange(1.5,1.5,0.01) y1 = x1**3  3*x1 plt.plot(x1,y1) plt.fill_between(x1, y1, 0, alpha=0.20) plt.plot([1,1],[2,2],'ko') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show() x2 = np.arange(3,3,0.01) def f(x): return x**3  3*x y2 = f(x2) plt.plot(x2,y2) plt.fill_between(x2, y2, 0, alpha=0.20) plt.plot([3,3],[f(3),f(3)],'ko') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show()
The condition that the function is defined on a closed interval is a crucial requirement for the extreme value theorem to hold true. Here is a counter example if this condition is relaxed. Let defined on the open interval . The function is unbounded; it keeps increasing as approaches . The figure below provides the plot of the function defined on the open interval . The function precipituously increases as it approaches the value of .
Rolle’s Theorem
Statement: Let be differentiable. Assume that , then there is at least one point where .
View Proof of Rolle's Theorem
The Extreme Value Theorem ensures that there is a local maximum or local minimum within the interval, while proposition 1 ensures that at this local extremum, the slope of the function is equal to zero. As an example, consider the function defined as . . This ensures that there is a point with . Indeed, and the point is the location of the local minimum. The following figure shows the graph of the function on the specified interval along with the point .
View Mathematica Code that Generated the Above FigureClear[x] y = 20 (x  1/2)^3  20 (x  1/2) + 5; Expand[y] y /. x > 1.5 y /. x > 0.5 y /. x > (1/2 + 1/Sqrt[3]) D[y, x] /. x > (1/2 + 1/Sqrt[3]) Plot[y, {x, 0.5, 1.5}, Epilog > {PointSize[0.04], Point[{1/2 + 1/Sqrt[3], y /. x > 1/2 + 1/Sqrt[3]}], Red, Line[{{3, y /. x > 1/2 + 1/Sqrt[3]}, {1.5, y /. x > 1/2 + 1/Sqrt[3]}}]}, Filling > Axis, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}]
import math import numpy as np import sympy as sp import matplotlib.pyplot as plt def f(x): return 20*(x  1/2)**3  20*(x  1/2) + 5 print("y(1.5):",f(1.5)) print("y(0.5):",f(0.5)) print("y(1/2 + 1/math.sqrt(3)):",f(1/2 + 1/math.sqrt(3))) x1 = sp.symbols('x') print("dy/dx(1/2 + 1/math.sqrt(3)):",sp.diff(20*(x1  1/2)**3  20*(x1  1/2) + 5,x1).subs(x1,1/2 + 1/math.sqrt(3))) x = np.arange(0.5,1.5,0.01) y = 20*(x  1/2)**3  20*(x  1/2) + 5 plt.plot(x,y) plt.fill_between(x, y, 0, alpha=0.20) plt.plot([1/2 + 1/math.sqrt(3)],[f(1/2 + 1/math.sqrt(3))],'ko') plt.plot([0.5,1.5],[f(1/2 + 1/math.sqrt(3)),f(1/2 + 1/math.sqrt(3))],'r') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show()
Generalized Rolle’s Thoerem
Statement: Let be times differentiable. Assume that is equal to zero at distinct points , then there is at least one point where .
View Proof of the Generalized Rolle's Theorem
Mean Value Theorem
Statement: Let be differentiable. Then, there is at least one point such that .
View Proof of Mean Value Theorem
Clearly, satisfies the conditions of Rolle’s theorem (Differentiable and ). We also have:
Therefore, by Rolle’s theorem, there is a point such that .
The mean value theorem states that there is a point inside the interval such that the slope of the function at is equal to the average slope along the interval. The following example will serve to illustrate the main concept of the mean value theorem. Consider the function defined as:
The slope or first derivative of is given by:
The average slope of on the interval is given by:
The two points and have a slope equal to the average slope:
The figure below shows the function on the specified interval. The line representing the average slope is shown in black connecting the points and . The red lines show the slopes at the points and .
View Mathematica Code that Generated the Above FigureClear[x] y = x^3  3 x; averageslope = ((y /. x > 3)  (y /. x > 3))/(3 + 3) dydx = D[y, x]; a = Solve[D[y, x] == averageslope, x] Point1 = {x /. a[[1, 1]], y /. a[[1, 1]]} Point2 = {x /. a[[2, 1]], y /. a[[2, 1]]} Plot[y, {x, 3, 3}, Epilog > {PointSize[0.04], Point[{3, y /. x > 3}], Point[{3, y /. x > 3}], Line[{{3, y /. x > 3}, {3, y /. x > 3}}], Point[Point1],Point[Point2], Red, Line[{Point1 + {1, averageslope}, Point1, Point1 + {1, averageslope}}], Line[{Point2 + {1, averageslope}, Point2, Point2 + {1, averageslope}}]}, Filling > Axis, PlotRange > All, Frame > True, AxesLabel > {"x", "y"}]
import numpy as np import sympy as sp import matplotlib.pyplot as plt def f(x): return x**3  3*x averageSlope = (f(3)  f(3))/(3 + 3) print("averageSlope:",averageSlope) x1 = sp.symbols('x') dydx = sp.diff(x1**3  3*x1,x1) print("dy/dx:",dydx) sol = list(sp.solveset(dydx  averageSlope,x1)) print("Solve:",sol) Point1 = [sol[0], f(sol[0])] Point2 = [sol[1], f(sol[1])] print("Point1:",Point1) print("Point2:",Point2) x = np.arange(3,3,0.01) y = x**3  3*x plt.plot(x,y) plt.fill_between(x, y, 0, alpha=0.20) plt.plot([3,3,Point1[0],Point2[0]],[f(3),f(3),Point1[1],Point2[1]],'ko') plt.plot([Point1[0]1,Point1[0]+1,Point1[0]], [Point1[1]averageSlope,Point1[1]+averageSlope,Point1[1]],'r') plt.plot([Point2[0]1,Point2[0]+1,Point2[0]], [Point2[1]averageSlope,Point2[1]+averageSlope,Point2[1]],'r') plt.plot([3,3],[f(3),f(3)],'k') plt.xlabel('x'); plt.ylabel('y') plt.grid(); plt.show()
First and Second Derviative Tests
The Mean Value Theorem precipitates two important results that are fundamental to analyze the behaviour of functions around their extreme values. First, we define the notion of increasing and decreasing functions
Definition: Increasing and Decreasing Functions
Let , then:
 is increasing if in
 is decreasing if in
A function is stricly increasing or decreasing if or respectively.
Proposition 2: First Derivative Test
Let be a continuous and smooth function. Then:
 If then is increasing
 If then is decreasing
 If then is constant
View Proof of the First Derivative Test
As , therefore implying that . Therefore, is increasing. Similar arguments follow for the second case and third cases.
Proposition 3: Second Derivative Test
Let be a continuous and smooth function. Let be such that . Then:
 If then is a local maximum
 If then is a local minimum
View Proof of the Second Derivative Test
But , therefore, . Since is arbitrary, the function is increasing on the interval . Similarly, let . Using the mean value theorem such that:
But , therefore, . Since is arbitrary, using Proposition 2 the function is decreasing on the interval . I.e., the function is increasing on the left of and decreasing on the right of . Therefore, is a local maximum. Similar arguments apply for the second case.
This proposition is very important for optimization problems when a local maximum or minimum is to be obtained for a particular function. In order to identify whether the solution corresponds to a local maximum or minimum, the second derivative of the function can be evaluated. Considering the example given above under Proposition 1, the second derivative is given by:
We have already identified and as locations of the local extremum values. To know whether they are local maxima or local minima, we can evaluate the second derivative at these points. . Therefore, is the location of a local minimum, while implying that is the location of a local maximum.
Taylor’s Theorem
As an introduction to Taylor’s Theorem, let’s assume that we have a function that can be represented as a polynomial function in the following form:
where is a fixed point and is a constant. The best way to find these constants is to find and its derivatives when . So, when we have:
Therefore, .
The derivatives of have the form:
The derivatives of when have the form:
Therefore, :
The above does not really serve as a rigorous proof for Taylor’s Theorem but rather an illustration that if an infinitely differentiable function can be represented as the sum of an infinite number of polynomial terms, then, the Taylor series form of a function defined at the beginning of this section is obtained. The following is the exact statement of Taylor’s Theorem:
Statement of Taylor’s Theorem: Let be times differentiable on an open interval . Let . Then, between and such that:
There are many proofs that can be found online for Taylor’s Theorem. Fundamentally, all of them rely on the Mean Value Theorem. We provide one proof in the expandable box below.
View Proof of Taylor's Theorem
Notice that substituting with gives:
The derivative of with respect to its variable gives:
Define a new function of given by:
The derivative of with respect to gives:
Evaluating the function at the points and gives:
Using the Mean Value Theorem, such that:
Rearranging:
Which is the difference between and the polynomial expantion around the point .
Explanation and Importance: Taylor’s Theorem has numerous implications in analysis in engineering. In the following we will discuss the meaning of the theorem and some of its implications:

Simply put, Taylor’s Theorem states the following: if the function and its derivatives are known at a point , then, the function at a point away from can be approximated by the value of the Taylor’s approximation :
The error (difference between the approximation and the exact is given by:
The term is bounded since is a continuous function on the interval from to . Therefore, when , the upper bound of the error can be given as:
While, when , the upper bound of the error can be given as:
The above implies that the error is directly proportional to . This is traditionally written as follows:where . In other words, as gets smaller and smaller, the error gets smaller in proportion to . As an example, if we choose and then , then, . I.e., if the step size is halved, the error is divided by .

If the function is infinitely differentiable on an interval , and if , then is the limit of the sum of the Taylor series. The error which is the difference between the infinite sum and the approximation is called the truncation error as defined in the error section.

There are many rigorous proofs available for Taylor’s Theorem and the majority rely on the mean value theorem above. Notice that if we choose , then the mean value theorem is obtained. For a rigorous proof, you can check one of these links: link 1 or link 2. Note that these proofs rely on the mean value theorem. In particular, L’HÃ´pital’s rule was used in the Wikipedia proof which in turn relies on the mean value theorem.
The following code illustrates the difference between the function and the Taylor’s polynomial . You can download the code, change the function, the point , and the range of the plot to see how the Taylor series of other functions behave.
View Mathematica CodeTaylor[y_, x_, a_, n_] := (y /. x > a) + Sum[(D[y, {x, i}] /. x > a)/i!*(x  a)^i, {i, 1, n}] f = Sin[x] + 0.01 x^2; Manipulate[ s = Taylor[f, x, 1, nn]; Grid[{{Plot[{f, s}, {x, 10, 10}, PlotLabel > "f(x)=Sin[x]+0.01x^2", PlotLegends > {"f(x)", "P(x)"}, PlotRange > {{10, 10}, {6, 30}}, ImageSize > Medium]}, {Expand[s]}}], {nn, 1, 30, 1}]
import math import numpy as np import sympy as sp import matplotlib.pyplot as plt from ipywidgets.widgets import interact def taylor(f,xi,a,n): return sum([(f.diff(x1,i).subs(x1,a))/math.factorial(i)*(xi  a)**i for i in range(n)]) x1 = sp.symbols('x') f = sp.sin(x1) + 0.01*x1**2 @interact(n=(1,30,1)) def update(n=1): x = np.arange(10,10,0.1) y = np.sin(x) + 0.01*x**2 plt.plot(x,y, label="f(x)") p = [taylor(f,xi,1,n) for xi in x] plt.plot(x,p, label="P(x)") plt.title("") plt.xlabel('x'); plt.ylabel('y') plt.ylim(6,30); plt.xlim(10,10) plt.legend(); plt.grid(); plt.show() print(sp.series(f,x1,0,n))
The following tool illustrates the difference between the function and the Taylor’s polynomial . You can change the order of the series expansion to see how the Taylor series of the function behave.
The Mathematica function Series can also be used to generate the Taylor expansion of any function:
Series[Tan[x],{x,0,7}] Series[1/(1+x^2),{x,0,10}]
import sympy as sp sp.init_printing(use_latex=True) x = sp.symbols('x') display("tan(x):",sp.series(sp.tan(x),x,0,8)) display("1/(1+x**2):",sp.series(1/(1+x**2),x,0,11))
The following tool shows how the Taylor series expansion around the point , termed in the figure, can be used to provide an approximation of different orders to a cubic polynomial, termed in the figure. Use the buttons to change the order of the series expansion. The tool provides the error at , namely . What happens when the order reaches 3?
Polynomial Interpolation Error
While not related to the Taylor’s Theorem, the error in the interpolating polynomial can be shown to have a form similar to the Taylor’s Theorem error term. The following theorem will be used later in the book when evaluating the error associated with the interpolating polynomial. Similar to Taylor’s Theorem, the proof relies on the Mean Value Theorem above.
Statement of Polynomial Interpolation Error Theorem: Let be times differentiable on an open interval . Let and define the degree interpolating polynomial
Then, between and such that:
View Proof of Polynomial Interpolation Error Theorem
Notice that as the interpolating polynomial coefficients are obtained by solving the equations of the form where . Fix such that and define the function as:
Where
Utilizing the fact that , the derivative of is given by:
Notice that and . I.e., has distinct roots. Using the generalized Rolle’s theorem, such that . Therefore:
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