Incalculus,Taylor's theorem gives an approximation of a-timesdifferentiable function around a given point by apolynomial of degree, called the-th-orderTaylor polynomial. For asmooth function, the Taylor polynomial is the truncation at the order of theTaylor series of the function. The first-order Taylor polynomial is thelinear approximation of the function, and the second-order Taylor polynomial is often referred to as thequadratic approximation.[1] There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
Taylor's theorem is named afterBrook Taylor, who stated a version of it in 1715,[2] although an earlier version of the result was already mentioned in1671 byJames Gregory.[3]
is the linear approximation of forx near the pointa, whose graph is thetangent line to the graph atx =a. The error in the approximation is:
Asx tends to a, this error goes to zero much faster than, making a useful approximation.
Graph of (blue) with its quadratic approximation (red) at. Note the improvement in the approximation.
For a better approximation to, we can fit aquadratic polynomial instead of a linear function:
Instead of just matching one derivative of at, this polynomial has the same first and second derivatives, as is evident upon differentiation.
Taylor's theorem ensures that thequadratic approximation is, in a sufficiently small neighborhood of, more accurate than the linear approximation. Specifically,
Here the error in the approximation is
which, given the limiting behavior of, goes to zero faster than asx tends to a.
Approximation of (blue) by its Taylor polynomials of order centered at (red) and (green). The approximations do not improve at all outside and, respectively.
Similarly, we might get still better approximations tof if we usepolynomials of higher degree, since then we can match even more derivatives withf at the selected base point.
In general, the error in approximating a function by a polynomial of degreek will go to zero much faster than asx tends to a. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to beanalytic atx = a: it is not (locally) determined by its derivatives at this point.
Taylor's theorem is of asymptotic nature: it only tells us that the error in anapproximation by a-th order Taylor polynomialPk tends to zero faster than any nonzero-th degreepolynomial as. It does not tell us how large the error is in any concreteneighborhood of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions onf. These enhanced versions of Taylor's theorem typically lead touniform estimates for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the functionf isanalytic. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)
There are several ways we might use the remainder term:
Estimate the error for a polynomialPk(x) of degreek estimating on a given interval (a –r,a +r). (Given the interval and degree, we find the error.)
Find the smallest degreek for which the polynomialPk(x) approximates to within a given error tolerance on a given interval (a −r,a +r) . (Given the interval and error tolerance, we find the degree.)
Find the largest interval (a −r,a +r) on whichPk(x) approximates to within a given error tolerance. (Given the degree and error tolerance, we find the interval.)
The polynomial appearing in Taylor's theorem is the-th order Taylor polynomial
of the function at the point. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if there exists a function and a-th order polynomialp such that
then. Taylor's theorem describes the asymptotic behavior of theremainder term
which is theapproximation error when approximatingf with its Taylor polynomial. Using thelittle-o notation, the statement in Taylor's theorem reads as
Under stronger regularity assumptions onf there are several precise formulas for the remainder termRk of the Taylor polynomial, the most common ones being the following.
for some real number between and. This is theLagrange form[8] of the remainder.
Similarly,
for some real number between and. This is theCauchy form[9] of the remainder.
Both can be thought of as specific cases of the following result: Consider
for some real number between and. This is theSchlömilch form of the remainder (sometimes called theSchlömilch-Roche). The choice is the Lagrange form, whilst the choice is the Cauchy form.
These refinements of Taylor's theorem are usually proved using themean value theorem, whence the name. Additionally, notice that this is precisely themean value theorem when. Also other similar expressions can be found. For example, ifG(t) is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between and, then
for some number between and. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below usingCauchy's mean value theorem. The Lagrange form is obtained by taking and the Cauchy form is obtained by taking.
The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding ofLebesgue integration theory for the full generality. However, it holds also in the sense ofRiemann integral provided the (k + 1)th derivative off is continuous on the closed interval [a,x].
It is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it. Suppose thatf is(k + 1)-times continuously differentiable in an intervalI containinga. Suppose that there are real constantsq andQ such that
throughoutI. Then the remainder term satisfies the inequality[11]
ifx >a, and a similar estimate ifx <a. This is a simple consequence of the Lagrange form of the remainder. In particular, if
on an intervalI = (a −r,a +r) with some , then
for allx∈(a −r,a +r). The second inequality is called auniform estimate, because it holds uniformly for allx on the interval(a −r,a +r).
Approximation of (blue) by its Taylor polynomials of order centered at (red).
Suppose that we wish to find the approximate value of the function on the interval while ensuring that the error in the approximation is no more than 10−5. In this example we pretend that we only know the following properties of the exponential function:
★
From these properties it follows that for all, and in particular,. Hence the-th order Taylor polynomial of at and its remainder term in the Lagrange form are given by
where is some number between 0 andx. Sinceex is increasing by (★), we can simply use for to estimate the remainder on the subinterval. To obtain an upper bound for the remainder on, we use the property for to estimate
using the second order Taylor expansion. Then we solve forex to deduce that
simply by maximizing thenumerator and minimizing thedenominator. Combining these estimates forex we see that
so the required precision is certainly reached, when
(Seefactorial or compute by hand the values and.) As a conclusion, Taylor's theorem leads to the approximation
For instance, this approximation provides adecimal expression, correct up to five decimal places.
LetI ⊂R be anopen interval. By definition, a functionf :I →R isreal analytic if it is locally defined by a convergentpower series. This means that for everya ∈ I there exists somer > 0 and a sequence of coefficientsck ∈ R such that(a −r,a +r) ⊂I and
This result is based on comparison with ageometric series, and the same method shows that if the power series based ona converges for someb ∈R, it must convergeuniformly on theclosed interval, where. Here only the convergence of the power series is considered, and it might well be that(a −R,a +R) extends beyond the domainI off.
The Taylor polynomials of the real analytic functionf ata are simply the finite truncations
of its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions
Here the functions
are also analytic, since their defining power series have the same radius of convergence as the original series. Assuming that[a −r,a +r] ⊂I andr < R, all these series converge uniformly on(a −r,a +r). Naturally, in the case of analytic functions one can estimate the remainder term by the tail of the sequence of the derivativesf′(a) at the center of the expansion, but usingcomplex analysis also another possibility arises, which is describedbelow.
The Taylor series off will converge in some interval in which all its derivatives are bounded and do not grow too fast ask goes to infinity. (However, even if the Taylor series converges, it might not converge tof, as explained below;f is then said to be non-analytic.)
One might think of the Taylor series
of an infinitely many times differentiable functionf :R →R as its "infinite order Taylor polynomial" ata. Now theestimates for the remainder imply that if, for anyr, the derivatives off are bounded over (a − r,a + r), then for any orderk and for anyr > 0 there exists a constantMk,r > 0 such that
★★
for everyx ∈ (a − r,a + r). Sometimes the constantsMk,r can be chosen in such way thatMk,r is bounded above, for fixedr and allk. Then the Taylor series offconverges uniformly to some analytic function
(One also gets convergence even ifMk,r is not bounded above as long as it grows slowly enough.)
The limit functionTf is by definition always analytic, but it is not necessarily equal to the original functionf, even iff is infinitely differentiable. In this case, we sayf is anon-analytic smooth function, for example aflat function:
for some polynomialpk of degree 2(k − 1). The function tends to zero faster than any polynomial as, sof is infinitely many times differentiable andf(k)(0) = 0 for every positive integerk. The above results all hold in this case:
The Taylor series off converges uniformly to the zero functionTf(x) = 0, which is analytic with all coefficients equal to zero.
The functionf is unequal to this Taylor series, and hence non-analytic.
For any orderk ∈ N and radiusr > 0 there existsMk,r > 0 satisfying the remainder bound (★★) above.
However, ask increases for fixedr, the value ofMk,r grows more quickly thanrk, and the error does not go to zero.
Letr > 0 such that theclosed diskB(z, r) ∪ S(z, r) is contained inU. Then Cauchy's integral formula with a positive parametrizationγ(t) =z +reit of the circleS(z,r) with gives
Here all the integrands are continuous on thecircleS(z, r), which justifies differentiation under the integral sign. In particular, iff is oncecomplex differentiable on the open setU, then it is actually infinitely many timescomplex differentiable onU. One also obtainsCauchy's estimate[12]
for anyz ∈ U andr > 0 such thatB(z, r) ∪ S(c, r) ⊂ U. The estimate implies that thecomplexTaylor series
off converges uniformly on anyopen disk with into some functionTf. Furthermore, using thecontour integral formulas for the derivativesf(k)(c),
so anycomplex differentiable functionf in an open setU ⊂ C is in factcomplex analytic. All that is said for real analytic functionshere holds also for complex analytic functions with the open intervalI replaced by an open subsetU ∈ C anda-centered intervals (a − r, a + r) replaced byc-centered disksB(c, r). In particular, the Taylor expansion holds in the form
where the remainder termRk is complex analytic. Methods of complex analysis provide some powerful results regarding Taylor expansions. For example, using Cauchy's integral formula for any positively orientedJordan curve which parametrizes the boundary of a region, one obtains expressions for the derivativesf(j)(c) as above, and modifying slightly the computation forTf(z) =f(z), one arrives at the exact formula
The important feature here is that the quality of the approximation by a Taylor polynomial on the region is dominated by the values of the functionf itself on the boundary. Similarly, applying Cauchy's estimates to the series expression for the remainder, one obtains the uniform estimates
Complex plot of. Modulus is shown by elevation and argument by coloring: cyan = , blue = , violet = , red = , yellow = , green = .
The function
isreal analytic, that is, locally determined by its Taylor series. This function was plottedabove to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. This kind of behavior is easily understood in the framework of complex analysis. Namely, the functionf extends into ameromorphic function
on the compactified complex plane. It has simple poles at and, and it is analytic elsewhere. Now its Taylor series centered atz0 converges on any discB(z0,r) withr < |z − z0|, where the same Taylor series converges atz ∈ C. Therefore, Taylor series off centered at 0 converges onB(0, 1) and it does not converge for anyz ∈C with |z| > 1 due to the poles ati and −i. For the same reason the Taylor series off centered at 1 converges on and does not converge for anyz ∈ C with.
If this is the case, then is the (uniquely defined)differential off at the pointa. Furthermore, then thepartial derivatives off exist ata and the differential off ata is given by
forα ∈Nn andx ∈Rn. If all the-th orderpartial derivatives off :Rn →R are continuous ata ∈Rn, then byClairaut's theorem, one can change the order of mixed derivatives ata, so the short-hand notation
for the higher orderpartial derivatives is justified in this situation. The same is true if all the (k − 1)-th order partial derivatives off exist in some neighborhood ofa and are differentiable ata.[13] Then we say thatf isktimes differentiable at the point a.
Using notations of the preceding section, one has the following theorem.
Multivariate version of Taylor's theorem[14]—Letf :Rn →R be ak-timescontinuously differentiable function at the pointa ∈Rn. Then there exist functionshα :Rn →R, where such that
The proof here is based on repeated application ofL'Hôpital's rule. Note that, for each,. Hence each of the first derivatives of the numerator in vanishes at, and the same is true of the denominator. Also, since the condition that the function be times differentiable at a point requires differentiability up to order in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its derivatives are differentiable in a neighborhood of. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified. So
where the second-to-last equality follows by the definition of the derivative at.
Alternate proof for Taylor's theorem in one real variable
LetG be any real-valued function, continuous on the closed interval between and and differentiable with a non-vanishing derivative on the open interval between and, and define
for some on the open interval between and. Note that here the numerator is exactly the remainder of the Taylor polynomial for. Compute
plug it into (★★★) and rearrange terms to find that
This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form.The Lagrange form of the remainder is found by choosing and the Cauchy form by choosing.
Remark. Using this method one can also recover the integral form of the remainder by choosing
but the requirements forf needed for the use of mean value theorem are too strong, if one aims to prove the claim in the case thatf(k) is onlyabsolutely continuous. However, if one usesRiemann integral instead ofLebesgue integral, the assumptions cannot be weakened.
Now we canintegrate by parts and use the fundamental theorem of calculus again to see that
which is exactly Taylor's theorem with remainder in the integral form in the case. The general statement is proved usinginduction. Suppose that
eq1
Integrating the remainder term by parts we arrive at
Substituting this into the formulain (eq1) shows that if it holds for the value, it must also hold for the value. Therefore, since it holds for, it must hold for every positive integer.
Derivation for the remainder of multivariate Taylor polynomials
We prove the special case, where has continuous partial derivatives up to the order in some closed ball with center. The strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of to the line segment adjoining and.[17] Parametrize the line segment between and by We apply the one-variable version of Taylor's theorem to the function:
Applying thechain rule for several variables gives
^Taylor, Brook (1715).Methodus Incrementorum Directa et Inversa [Direct and Reverse Methods of Incrementation] (in Latin). London. p. 21–23 (Prop. VII, Thm. 3, Cor. 2). Translated into English inStruik, D. J. (1969).A Source Book in Mathematics 1200–1800. Cambridge, Massachusetts: Harvard University Press. pp. 329–332.
^Genocchi, Angelo; Peano, Giuseppe (1884),Calcolo differenziale e principii di calcolo integrale, (N. 67, pp. XVII–XIX):Fratelli Bocca ed.{{citation}}: CS1 maint: location (link)
^This follows from iterated application of the theorem that if the partial derivatives of a functionf exist in a neighborhood ofa and are continuous ata, then the function is differentiable ata. See, for instance,Apostol 1974, Theorem 12.11.
