Chapter VI.
Taylor's Theorem and the Approximate Expression of Functions by Polynomials
In many respects, the rational functions are the simplest functions of analysis. They are formed by a finite number of applications of the rational operations of calculation, while in the last resort the formation of every other function involves a more or less concealed passage to the limit from rational functions. The questions whether and how a given function can be expressed approximately by rational functions, in particular, by polynomials are therefore of great importance in theory as well as in practice.
6.1 The Logarithm and the Inverse Tangent
6.1.1 The Logarithm: We begin with some special cases in which the integration of the geometrical progression leads almost at once to the desired approximations. We first remind the reader of the following fact. For q ¹ 1 and for positive integers n, we have
where
If q < 1, the remainder rntends to 0 as n increases and we then obtain (cf. 1.5.7) the infinite geometric series
We take as our starting-point the formula
and expand the integrand in accordance with the above formula, setting q = -t. Then we obtain at once by integration
where
Hence, for any positive integer n, we have expressed the function log (1 + x) approximately by a polynomial of the n-th degree, namely
at the same time, the quantity Rn - the remainder - specifies the size of the error made in the approximation.
In order to estimate the accuracy of this approximation, we need only have an estimate for the remainder Rn; such an estimate is given to us immediately by the integral estimates in 2.7. If we assume at first that x ³ 0, then the integrand is nowhere negative in the entire interval of integration and nowhere exceeds tn. Consequently,
and we therefore see that, for every value of x in the interval 0 £ x £ 1, this remainder can be made as small as we please by choosing n large enough (cf. 1.4). If, on the other hand, the quantity x is in the interval -1 < x £ 0, the integrand will not change sign and its absolute value will not exceed |t|n/(l + x), and we thus obtain the remainder estimate
Hence, we see that here again the remainder is arbitrarily small when we make n sufficiently large. Of course, our estimate has no meaning for x = -1.
Summing up, we can say that
where the remainder Rn tends to zero as n increases, provided that x lies in the interval -1 < x £ 0; note that this interval is open on the left hand side and closed on the right hand side. In fact, we can find from the above inequalities an estimate for the remainder, independent of x, which is valid for all values of x in the interval -1 + h £ x £ 1, where h is a number such that 0 < h £ 1. In fact, then
and this formula shows that in the entire interval the function log (1 + x) is expressed approximately by our polynomial of the n-th degree, the error being nowhere greater than 1/h(n + 1). We leave it to the reader to convince himself that for all values of x for which |x| > 1, the remainder not only fails to approach zero, but, in fact, increases numerically beyond all bounds as n increases, so that for such values of x our polynomial does not yield an approximation to the logarithm.
The fact that in the above interval the remainder Rn tends to zero may be expressed by saying that we have in this interval for the logarithm the infinite series (cf. Chapter VIII)
If we insert in this series the particular value x = 1, we obtain the remarkable formula
This is one of the relations the discovery of which left a deep impression on the minds of the first pioneers of the differential and integral calculus.
The above approximation for the logarithm leads us to another formula which is useful for many purposes, particularly in numerical calculations. Provided that -1 < x < 1, we have only to write -x in place of x in the above formula to obtain
Assuming that n is even and subtracting, we find
where Rn is given by
Because
the remainder tends to zero as n increases, a fact which we again express by writing the expansion as an infinite series:
for all values of x such that |x| < 1.
An advantage of this formula is that as x traverses the interval -1 to +1, the expression (1 + x)/(1 - x) ranges over all positive numbers, whence, if the value of x is chosen suitably, this series enables us to calculate the value of the logarithm of any positive number with an error not exceeding the above estimate for R.
6.1.2 The Inverse Tangent: We can treat the inverse tangent in a similar way, if we begin with the formula, true for every positive integer n,
where
By integration, we obtain
and see at once that in the interval -1 £ x £ 1 the remainder Rn tends to zero as n increases, because, by the mean value theorem of the integral calculus,
Using this formula for the remainder, we can also readily show that for |x| > 1 the absolute value of the remainder increases beyond all bounds as n increases. We have thus derived the infinite series
valid for |x| £ 1. For x = 1, since arc tan 1 = p /4, we have
as remarkable a formula as the one found previously for log 2.
Exercises 6.1:
1. Prove that
Hence find log 4/3 to 2 places.
2. Calculate log 6/5 to 3 places, using the series
Prove that the result is accurate to 3 places.
3. How many terms of the series for log (1 + x} must be used in order to obtain log (1 + x) to within 10 per cent, if 30 £ x £ 31?
An approximate representation by rational functions, as in the special cases above, can also be obtained in the case of an arbitrary function f(x) for which we only assume that for all values of the independent variable in an assigned closed interval the function has continuous derivatives at least up to the (n+l)-th order . In most of the cases, which actually occur, the existence and continuity of all the derivatives of the function are known to begin with, so that we can choose for n any arbitrary integer. The approximation formula, which we shall now derive, was discovered in the early days of the differential and integral calculus by Newton's student Taylor and is known as Taylor's theorem.
A special case of this theorem is often referred to, without historical justification, as Maclaurin's theorem. We will not follow this practice.
6.2.1 Taylor's Theorem for Polynomials: In order to get a clear idea of the problem, we shall consider first the case where
is itself a polynomial of the n-th degree. We can then easily express the coefficients of this polynomial in terms of the derivatives of f(x) at the point x=0. In fact, differentiating both sides of the equation once, twice, etc., with respect to x and setting x = 0, we at once find that the coefficients are
Any polynomial f(x) of the n-th degree can therefore be written in the form
This formula merely states that the coefficients an can be expressed in terms of the derivatives at x = 0 and gives the expressions for them.
We can generalize this Taylor series for the polynomial slightly, if we replace x by x = x+ h and consider the function f(x) = f(x + h) =g (h) as a function of h, for moment thinking of x as fixed and h as the independent variable. It then follows that
and hence, if we set h = 0,
If we apply the previous formula to the function f(x + h) = g(h), which is itself a polynomial of the n-th degree in h, we immediately obtain the Taylor series
6.2.2 Taylor's Theorem for an Arbitrary Function: These formulae suggest that we should seek a similar formula in the case of an arbitrary function f(x), which is not necessarily a polynomial; however, in this case, the formula can lead only to an approximation to the function by a polynomial.
We wish to compare the values of the function f at the point x and at the point x=x+h, so that h = x - x. If now n is any positive integer, as a rule, the expression
will not be an exact expression for the functional value f(x), whence we must set
where the expression Rn
denotes the remainder when f(x) is replaced by the expression 
. In the first
instance, this equation is nothing but a formal definition of the
expression Rn. Its significance lies
in the fact that we can easily find a neat and useful expression
for Rn. For this purpose, we think
of the quantity x as the fixed and the quantity x as the
independent variable. The remainder is then the function Rn(x).
By the above equation, this function vanishes for x = x:
Moreover, by differentiation, we obtain
In fact, if we differentiate this equation for the remainder with respect to x, we obtain 0 on the left hand side, since f(x) does not depend on x and is therefore to be regarded as a constant. On the right hand side, we differentiate each term by the rule for products and find that all the terms cancel except the last one, which is written above with a minus sign.
Now, by the fundamental theorem of the integral calculus,
so that we obtain the formula
If we introduce a new integration variable t by means of the equation t =t-x, this becomes
Collecting these results, we can make the statement:
If the function f(x)
has continuous derivatives up to order (n + l) in the
interval under consideration, then 
or (the equivalent expression for h = x - x)
where the remainder Rn is given by
In particular, if we set x = 0 and then replace h by x, we obtain
with the remainder
These formulae are known as Taylor's theorem. They give expressions for the functions f(x + h) and f(x), respectively, in terms of polynomials of degree n in h and in x, respectively (the so-called polynomial of approximation), and a remainder. The polynomial of approximation is characterized by the fact that, when h = 0 (or x = 0, as the case may be), its value and that of its first n derivatives are the same as those of the given function and its first n derivatives. In contrast to the Taylor series for polynomials, which do not require a remainder, the remainder and the expression for it are here essential. The significance of the formula lies in the fact that the remainder, even though it has a more complicated form than the other terms of the formula, nevertheless provides useful means for estimating the accuracy with which the sum of the first n + 1 terms
represents the function f(x).
6.2.3 Estimation of the Remainder: Whether the first n + 1 terms of Taylor's series actually yield a sufficiently good approximation to the function depends naturally on whether the remainder is sufficiently small. Hence we turn now to the estimation of this remainder. Such an estimate can most easily be made by means of the mean value theorem of the integral calculus (2.7).
We employ this theorem in the form
where p(t) is a continuous function which is negative nowhere in the interval of integration and f(t) is merely a continuous function there, while q is a number in the interval 0 £ q £ l. In fact, we may assume that 0 < q < 1, but this is not important here. If we set in the expression for the remainder p(t)=(h-t), we obtain
while if we set p(t) = 1, we obtain
which is less important for us and is stated
here only for the sake of completeness. In these formulae, q denotes a
certain number in the interval
0 £ q £ l, the value of which, in general, cannot be specified
more accurately; of course, as a rule, this value is different in
the two formulae for the remainder and depends, in addition, on n,
x and h. The first form of the remainder was
given by Lagrange, the second by Cauchy,
whence they received their names.
These expressions for the remainder, as well as others, can be derived from the mean value theorem of the differential calculus and from the generalized mean value theorem (A3.3), respectively. We apply these theorems to the function Rn = Rn(x) - Rn(x) and to the pair of functions Rn(x) and (x - x)n+1, respectively, where we consider x to be fixed and employ the formula
These methods of deriving the formulae for the remainder emphasize more the fact that Taylor's theorem is a generalization of the mean value theorem; they also offer the advantage, which for many theoretical purposes is important, that we need only assume the existence and not the continuity of the (n + l)-th derivative. However, on the other hand, we lose the advantage of having an exact expression for the remainder in the form of an integral.
Our interest will be directed chiefly towards discovering whether the remainder Rn tends to zero as n increases; if this is the case, the larger we choose n the more accurately is f(x + h) represented by the corresponding polynomial in h. In this case, we say that we have expanded the function in an infinite Taylor series
or, in particular, if we first set x = 0 and then write x in place of h,
We shall encounter examples in the next section. However, first of all, we wish to point out the second important point of view arising from the consideration of Taylor series. If we think in the first formula of the quantity h as becoming smaller and tending to zero, then, in the terminology of 3.9.5, the various terms of the series will tend to zero at different orders of magnitude; we accordingly call the expression f(x) the term of zero order in the Taylor series, the expression hf '(x) the term of first order, the expression h²f"(x)/2! the term of second order, etc. We gather from the form of our remainder the fact:
In expanding a function as far as the term of n-th order, we make an error which tends to zero at order (n +1) as h ® 0.
Many important applications depend on this fact. It shows us that the nearer the point x + h lies to the point x, the better is the representation of the function f(x+h) by the polynomial approximation and that, in a given case, the approximation can be improved in the immediate neighbourhood of the point x by increasing the value of n.
1. Let f(x) have a continuous derivative in the interval a £ x £ b and f "(x) ³ 0 for ³ every value of x. Then, if x is any point in the interval, the curve nowhere drops below its tangent at the point x = x, y = f(x). (Use the Taylor expansion with three terms.)
2. Find the value of q in Lagrange's form of the remainder Rn for the expansions of 1/(1 - x) and 1/(1 + x) in powers of x.
6.3 Applications. Expansions of the Elementary Functions
We shall now use the general results of the preceding section in order to express the elementary functions approximately by polynomials and to expand them in Taylor series. However, we shall retrict ourselves to those functions for which the coefficients of the expansion in series are given by simple laws of formation. The series for certain other functions will be discussed in 8.6.
6.3.1 The Exponential function: The simplest example is offered by the exponential function f(x) = ex, all the derivatives are which identical with f(x) and therefore have the value 1 for x = 0. Hence, using Lagrange's form for the remainder, we at once obtain
in accordance with 6.2. If we now let n increase beyond all bounds, the remainder will tend to zero, no matter what fixed value of x has been chosen, because |eqx| £ e|x| from the start. We now choose a fixed integer m large than 2|x|, then for n £ m
whence
Since the first two factors on the right hand side are independent of x, while the number 1/2n tends to zero as n increases, our statement is proved. If we think of the number x as not being fixed, but free to vary in the interval -a £ x £ a, where a is a fixed positive number, there follows from the above, if we choose m > 2a, that the estimate
is valid provided n ³ m. Thus, we have specified for the remainder a bound which holds for all values of x in the interval -a £ x £ a and tends to zero as n®¥. For the function ex, we can therefore write the expansion as an infinite series
This expansion is valid for all values of x. Thus, we have again proved that the number e considered in 1.6.5 is the same as the base of the natural logarithm (3.6). Of course, we must employ for numerical calculations the finite form of Taylor's theorem with the remainder; for example, for x =1, this yields
If we wish to compute e with an error of at most 1/10,000, we need only choose n so large that the remainder is certainly less than 1/10,000; since this remainder is certainly less* than 3/(n + 1)!, it is sufficient to choose n = 7, since 8!>30,000. We thus obtain the approximate value
e = 2.71822
with an error less than 0.0001. We do not take here into account the error due to neglect of the figures in the sixth decimal place.
* We have used here the fact that e < 3. This follows immediately from our series for e, because it is always true that 1/n £ 1/2n-1, whence
6.3.2 Sin x,
cos x, sinh x,
cosh x:
We find for the functions sin x, cos x,
sinh x and cosh x:
If f(x) = sin x or f(x) = cos x, the n-th derivative can always be represented by
Hence, the coefficients of the even powers of x vanish in the polynomial approximations for sin x and sinhx, while the coefficients of the odd powers vanish in those for cos x and cosh x. Thus, in the first case, the (2n + l)-th and the (2n + 2)-th polynomial are identical, while, in the second case, the 2n-th and the (2n + l)-th polynomials are identical. If we employ in each case the higher order ones of these polynomials, we obtain at once Lagrange's forms of the remainders
where, of course, in each of the four formulae q denotes a different number in the interval 0 £ q £ l, a number which, in addition, depends on n and on x. In these formulae, we can also make the approximation as exact as we please for each value of x, since the remainder tends to 0 as n increases. We thus obtain the four series
The last two series can also be obtained formally from the series for ex in accordance with the definitions of the hyperbolic functions.
6.3.3 The Binomial Series: We may pass over the Taylor series for the functions log (1 + x) and artan x, which we have already dealt with directly in 6.1.1. However, we must take up the generalization of the binomial theorem for arbitrary indices, which is one of the most fruitful of Newton's mathematical discoveries and which represents one of the most important cases of the expansion in Taylor series. Our objective is the expansion of the function
in a Taylor series, where x > -1 and a is an arbitrary number, positive or negative, rational or irrational. We have chosen the function (1 + x)a instead of xa, because it is not true at the point x = 0 that all the derivatives of xa are continuous except in the trivial case of non-negative values of a. We calculate first the derivatives of f(x):
In particular, we have for x = 0,
Taylor's theorem then yields
We must still discuss the remainder. This is not very difficult, but nevertheless is not quite as simple as in the cases treated previously. Here we shall pass over the remainder estimate, since the general binomial theorem will be proved completely in a somewhat different and simpler way in 8.6.2 and A6.3. The result, which we mention here in advance, is that whenever |x| < 1, the remainder tends to 0, whence the expression (1 + x)a can be expanded in the infinite binomial series
where, for the sake brevity, we have introduced the general binomial coefficients
1. Expand (1+ x)1/2 into two terms plus remainder. Estimate the remainder.
2. Use the expansion of 1. (discarding the
remainder) to calculate 
What is the degree of accuracy of the
approximation?
3.. What linear function approximates best 
in the
neighbourhood of x = 0? Between what values of x
is the error of the approximation less than .01?
4. What quadratic function approximates best to ~(1 +
»} in the neighbourhood of x = 0? What is the greatest
error in the interval -0.1£ x£ 0.1?
5. (a) What linear function, (b)
what quadratic function approximates
best 
in the neighbourhood of x = 0? What are the largest
errors in -0.1£ x£ 0.1?
6. Calculate sin 0.01 to 4 places.
7. Do the sums for (a) cos 0.01, (b)
(c)
8. Expand sin (x + h) in a Taylor series in h, whence find sin 31º=sin(30º+1º) to 3 places.
Expand the functions in 9. - 18. in the neighbourhood of x = 0 to three terms plus remainder (writing the remainder in Lagrange's form).
19. (a) Expand esin x to five terms plus remainder; (b) in the power series for ez, substitute for z the power series for sin x, taking enough terms to secure that the coefficient of x4 is correct. Compare with (a).
20. Find the polynomial of fourth degree which best approximates tan x in the neighbourhood of * = 0. In what interval does this polynomial represent tan x to within 5%?
21. Find the first 6 terms of the Taylor series for y in powers of x for the functions, defined by
The behaviour of a function f(x) in the neighbourhood of a point x = a or of a given curve in the neighbourhood of a point can be studied with increased accuracy by means of Tailor's theorem, because this theorem resolves the increment of the function on passing to a neighbouring point x = a + h into a sum of quantities of the first order, second order, etc.
6.4.1 Contact of Curves: We shall now use this method, in order to investigate the concept of contact of two curves. If at a point, say the point x = a, two curves y = f(x) and y = g(x) do not only intersect, but also have a common tangent, we shall say that at this point the curves touch one another or have contact of the first order. The Taylor expansions of the functions f(a + h) and g(a + h) then have the same terms of zero and first order in h. If at the point x=a the second derivatives of f(x) and g(x) are also equal to each other, we say that the curves have contact of the second order. In the Taylor expansions, the terms of second order are then also the same and, if we assume that both functions have at least continuous derivatives of the third order, the difference D(x) = f(x) - g(x) can be expressed in the form
where the expression F(h) tends to f '"(a + h) - g"'(a) as h tends to zero, whence the difference D(a + h) vanishes to at least the third order in h.
We can proceed in this way and consider the general case, when the Taylor series for f(x) and g(x) are the same up to terms of the n-th order, i.e., when
We assume here that the (n + 1)-th derivatives are also continuous. Under these conditions, we say that at this point the curves have contact of the n-th order. The difference of the two functions will then be
where, since 0 £ q £ 1, as h tends to 0, the quantity F(h) = D(n+1)(a + q h) tends to f (n+1)(a) - g(n+1)(a). We recognize from this formula that at the point of contact the difference f(g) - g(x) vanishes at least to (n + l)-th order.
The Taylor polynomials are simply defined geometrically by the fact that they are those parabolas of order n which at the given point have contact of the highest possible order with the graph of the given function. Hence they are sometimes called osculating parabolas. For the case y = ex, Fig. 1 gives the first three osculating parabolas at the point x = 0.
If two curves y = f(x) and y = g(x) have contact of order n, the definition does not exclude the possibility that the contact may be of a still higher order, i.e., that also the equation f (n+1)(a) = g(n+1)(a) is true. If this is not the case, i.e., if f (n+1)(a) ¹ g(n+1)(a), we speak of a contact of exactly the n-th order or say that the order of the contact is exactly n.
The fact that the order of contact of two curves is a genuine geometrical relationship, which is unaffected by a change of axes, is easily verified by means of the formulae for a change of axes.
From our formula as well as from our figures, we can at once state a remarkable fact which is often overlooked by beginners. If the contact of two curves is exactly of an even order, that is, if an even number n of derivatives of the two functions have the same value at the point in question, while the (n+1)-th derivatives differ, then, in conformity with the above formula, the difference f (a + h) - g(a + h) will have different signs for small positive values and for numerically small negative values of h. The two curves will then cross at the point of contact. This case occurs, for example, in a contact of the second order, if the third derivatives have different values. However, if we consider in detail the case of an odd order contact, e.g., the case of an ordinary contact of the first order, the difference f (a + h) - g(a + h) will have the same sign for all numerically small values of h, whether positive or negative; the two curves therefore will not cross in the neighbourhood of the point of contact.
The simplest example of this type is the contact of a curve with its tangent. The tangent can cross the curve only at points where the contact is at least of second order; it will actually cross the curve at points, where the order of contact is even, e.g., at an ordinary point of inflection, where f "(x) = 0, but f'"(x) ¹ 0. At points, where the order of contact is odd, it will not cross the curve; examples are an ordinary point of the curve where the second derivative is non-zero or the curve y = x4 at the origin.
6.4.2 The Circle of Curvature as Osculating Circle:. From this point of view, the concept of the curvature of a curve y = f(x) gains a new intuitive significance. There pass through the definite point of the curve with co-ordinates x = a and y = b an infinite number of circles which touch the curve at the point. The centres of these circles lie on the normal to the curve and there corresponds to each point of this normal just one such tangent circle. We may expect that we can bring about by a proper choice of the centre of the circle a contact of second order between the curve and the circle.
As a matter of
fact, we know from 5.2.6 that for the circle of curvature at the point x
= a, the equation of which is, say, y =g
(x), we do not only have g(a)=f(a)
and g'(a) = f '(a), but also g"(a)
= f "(a), whence the circle of curvature
is at the same time the osculating circle at the point of the
curve under consideration, i.e., it is the circle which at that
point has second order contact with the curve. In the limiting
case of a point of inflection or, in general, of a point at which
the curvature is zero and the radius of curvature infinite, the
circle of curvature degenerates into the tangent. In ordinary
cases, i.e., when the contact at the point in question does not
happen to be of an order higher than the second, the circle of
curvature will not merely touch the curve, but will also cross it
(Fig. 2).
6.4.3 On the Theory of Maxima and Minima: As we have already seen in 3.5.2, a point x = a, at which f '(a) = 0, has a maximum of the function f(x) if f"(a) is negative, a minimum if f "(a) is positive,whence these conditions are sufficient for the occurrence of a maximum or a minimum. They are by no means necessary, because there are three possibilities in the case where f "(a)=0; at the point in question, the function may have a maximum or a minimum or neither. Examples of the three possibilities are giver by the functions y=-x4, y=x4 and y = x3 at the point x = 0. Taylor's theorem enables us immediately to make a general statement of sufficient conditions for a maximum or a minimum.We merely need expand the function f(a + h) in powers of h; the essential point is then to discover whether the first non-vanishing term contains an even or an odd power of h. In the first case, we have a maximum or a minimum according to whether the coefficient of h is negative or positive; in the second case, we have a horizontal inflectional tangent and neither a maximum nor a minimum. The reader should complete the argument by using the formula for the remainder.
However, the earlier necessary and sufficient condition (3.5.2) is more general and more convenient in applications, i.e., provided the first derivative f '(x) vanishes at only a finite number of points; a necessary and sufficient condition for the occurrence of a maximum or minimu,at one of these points is that the first derivative f '(x) changes sign as the point is passed.
1. What is the order of contact of the curves y=ex and y=l+x+½sin²x at x=0?
2. What is the order of contact of y = sin4x and y = tan4x at x = 0?
3. Determine the constants a, b, c, d in such a way that the curses y=e2x and y= acos x + b sin x + c cos 2x + d sin 2x have contact of order 3 at x = 0.
4. What is the order of contact of the curves
at their points of intersection? Plot the curves.
5. What is the order of contact of the curves
at their points of intersection?
6. The curve y = f(x) passes
through the origin O and touches the x-axis at
0. Show that the radius of curvature of the curve at O
is given by 
7.* E is a circle which touches a given curve at a point P and passes through a neighbouring point Q of the curve. Show that the limit of the circle K as Q®P is the circle of curvature of the curve at P.
8.* R is the point of intersection of the two normals to a given curve at the neighbouring points P, Q of the curve. Show that, as Q®P, R tends to the centre of curvature of the curve for the point P, (The centre of curvature is the intersection of neighbouring normals.)
9.* Show that the order of contact of a curve and the osculating circle is at least three at points where the radius of curvature is a maximum or a minimum.
10. Determine the maxima and minima of the function y = e-1/x² (cf. A.6).