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(5.73) is multiplied by |(1 −ααn+1 )x|. But since |x| < 1, we shall have |(1 − n+1 )x| < q < 1, independently ofthe value of α, provided |x| < q < 1 and n is sufficiently large.It follows from this that rn (0; x) → 0 as n → ∞ for any α ∈ R and any x inthe open interval |x| < 1. Therefore the expansion obtained by Newton (Newton’sbinomial theorem) is valid on the open interval |x| < 1:α(α − 1) 2αα(α − 1) · · · (α − n + 1) nx+x + ··· +x + ··· .1!2!n!(5.74)We remark that d’Alembert’s test (see Paragraph b of Sect. 3.1.4) implies that for|x| > 1 the series (5.74) generally diverges if α ∈/ N.

Let us now consider separatelythe case when α = n ∈ N.In this case f (x) = (1 + x)α = (1 + x)n is a polynomial of degree n and hence allof its derivatives of order higher than n are equal to 0. Therefore Taylor’s formula,together with, for example, the Lagrange form of the remainder, enables us to writethe following equality:(1 + x)α = 1 +n(n − 1) 2nn(n − 1) · · · 1 nx+x + ··· +x ,(5.75)1!2!n!which is the Newton binomial theorem known from high school for a naturalnumber exponent: nn nn 2(1 + x)n = 1 +x+x .x + ··· +1n2(1 + x)n = 1 +Thus we have defined Taylor’s formula (5.51) and obtained the forms (5.52),(5.55), and (5.56) for the remainder term in the formula. We have obtained the relations (5.58), (5.60), (5.62), (5.69), and (5.73), which enable us to estimate the errorin computing the important elementary functions using Taylor’s formula.

Finally,we have obtained the power-series expansions of these functions.Definition 6 If the function f (x) has derivatives of all orders n ∈ N at a point x0 ,the seriesf (x0 ) +1 1f (x0 )(x − x0 ) + · · · + f (n) (x0 )(x − x0 )n + · · ·1!n!is called the Taylor series of f at the point x0 .5.3 The Basic Theorems of Differential Calculus223It should not be thought that the Taylor series of an infinitely differentiable function converges in some neighborhood of x0 , for given any sequencec0 , c1 , . . .

, cn , . . . of numbers, one can construct (although this is not simple to do)a function f (x) such that f (n) (x0 ) = cn , n ∈ N.It should also not be thought that if the Taylor series converges, it necessarilyconverges to the function that generated it. A Taylor series converges to the functionthat generated it only when the generating function belongs to the class of so-calledanalytic functions.Here is Cauchy’s example of a nonanalytic function:!2e−1/x , if x = 0,f (x) =0,if x = 0.Starting from the definition of the derivative and the fact that x k e−1/x → 0 asx → 0, independently of the value of k (see Example 30 in Sect. 3.2), one can verifythat f (n) (0) = 0 for n = 0, 1, 2, .

. . . Thus, the Taylor series in this case has all itsterms equal to 0 and hence its sum is identically equal to 0, while f (x) = 0 if x = 0.In conclusion, we discuss a local version of Taylor’s formula.We return once again to the problem of the local representation of a functionf : E → R by a polynomial, which we began to discuss in Sect. 5.1.3. We wish tochoose the polynomial Pn (x0 ; x) = x0 + c1 (x − x0 ) + · · · + cn (x − x0 )n so as tohavef (x) = Pn (x) + o (x − x0 )n as x → x0 , x ∈ E,2or, in more detail,f (x) = c0 + c1 (x − x0 ) + · · · + cn (x − x0 )n + o (x − x0 )nas x → x0 , x ∈ E.(5.76)We now state explicitly a proposition that has already been proved in all its essentials.Proposition 3 If there exists a polynomial Pn (x0 ; x) = c0 + c1 (x − x0 ) + · · · +cn (x − x0 )n satisfying condition (5.76), that polynomial is unique.Proof Indeed, from relation (5.76) we obtain the coefficients of the polynomial successively and completely unambiguouslyc0 =c1 =lim f (x),Ex→x0limf (x) − c0,x − x0limf (x) − [c0 + · · · + cn−1 (x − x0 )n−1 ].(x − x0 )nEx→x0...cn =Ex→x0We now prove the local version of Taylor’s theorem.2245Differential CalculusProposition 4 (The local Taylor formula) Let E be a closed interval having x0 ∈ Ras an endpoint.

If the function f : E → R has derivatives f (x0 ), . . . , f (n) (x0 ) upto order n inclusive at the point x0 , then the following representation holds:f (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n +f (x) = f (x0 ) +1!n!+ o (x − x0 )n as x → x0 , x ∈ E.(5.77)Thus the problem of the local approximation of a differentiable function is solvedby the Taylor polynomial of the appropriate order.Since the Taylor polynomial Pn (x0 ; x) is constructed from the requirementthat its derivatives up to order n inclusive must coincide with the corresponding(k)derivatives of the function f at x0 , it follows that f (k) (x0 ) − Pn (x0 ; x0 ) = 0(k = 0, 1, . . . , n) and the validity of formula (5.77) is established by the followinglemma.Lemma 2 If a function ϕ : E → R, defined on a closed interval E with endpointx0 , is such that it has derivatives up to order n inclusive at x0 and ϕ(x0 ) = ϕ (x0 ) =· · · = ϕ (n) (x0 ) = 0, then ϕ(x) = o((x − x0 )n ) as x → x0 , x ∈ E.Proof For n = 1 the assertion follows from the definition of differentiability of thefunction ϕ at x0 , by virtue of whichϕ(x) = ϕ(x0 ) + ϕ (x0 )(x − x0 ) + o(x − x0 )as x → x0 , x ∈ E,and, since ϕ(x0 ) = ϕ (x0 ) = 0, we haveϕ(x) = o(x − x0 )as x → x0 , x ∈ E.Suppose the assertion has been proved for order n = k − 1 ≥ 1.

We shall showthat it is then valid for order n = k ≥ 2.We make the preliminary remark that sinceϕ (k) (x0 ) = ϕ (k−1) (x0 ) =limEx→x0ϕ (k−1) (x) − ϕ (k−1) (x0 ),x − x0the existence of ϕ (k) (x0 ) presumes that the function ϕ (k−1) (x) is defined on E, atleast near the point x0 .

Shrinking the closed interval E if necessary, we can assumefrom the outset that the functions ϕ(x), ϕ (x), . . . , ϕ (k−1) (x), where k ≥ 2, are alldefined on the whole closed interval E with endpoint x0 . Since k ≥ 2, the functionϕ(x) has a derivative ϕ (x) on E, and by hypothesis (k−1) (x0 ) = 0.ϕ (x0 ) = · · · = ϕ Therefore, by the induction assumption,ϕ (x) = o (x − x0 )k−1 as x → x0 , x ∈ E.5.3 The Basic Theorems of Differential Calculus225Then, using Lagrange’s theorem, we obtainϕ(x) = ϕ(x) − ϕ(x0 ) = ϕ (ξ )(x − x0 ) = α(ξ )(ξ − x0 )(k−1) (x − x0 ),where ξ lies between x0 and x, that is, |ξ − x0 | < |x − x0 |, and α(ξ ) → 0 as ξ →x0 , ξ ∈ E.

Hence as x → x0 , x ∈ E, we have simultaneously ξ → x0 , ξ ∈ E, andα(ξ ) → 0. Since ϕ(x) ≤ α(ξ )|x − x0 |k−1 |x − x0 |,we have verified thatϕ(x) = o (x − x0 )k as x → x0 , x ∈ E.Thus, the assertion of Lemma 2 has been verified by mathematical induction.Relation (5.77) is called the local Taylor formula since the form of the remainderterm given in it (the so-called Peano form)rn (x0 ; x) = o (x − x0 )n ,(5.78)makes it possible to draw inferences only about the asymptotic nature of the connection between the Taylor polynomial and the function as x → x0 , x ∈ E.Formula (5.77) is therefore convenient in computing limits and describing theasymptotic behavior of a function as x → x0 , x ∈ E, but it cannot help with theapproximate computation of the values of the function until some actual estimate ofthe quantity rn (x0 ; x) = o((x − x0 )n ) is available.Let us now summarize our results.

We have defined the Taylor polynomialPn (x0 ; x) = f (x0 ) +f (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n ,1!n!written the Taylor formulaf (x) = f (x0 ) +f (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n + rn (x0 ; x),1!n!and obtained the following very important specific form of it:If f has a derivative of order n + 1 on the open interval with endpoints x0 and x,thenf (x) = f (x0 ) ++f (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n +1!n!f (n+1) (ξ )(x − x0 )n+1 ,(n + 1)!where ξ is a point between x0 and x.(5.79)2265Differential CalculusIf f has derivatives of orders up to n ≥ 1 inclusive at the point x0 , thenf (x) = f (x0 ) +f (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n + o (x − x0 )n . (5.80)1!n!Relation (5.79), called Taylor’s formula with the Lagrange form of the remainder,is obviously a generalization of Lagrange’s mean-value theorem, to which it reduceswhen n = 0.Relation (5.80), called Taylor’s formula with the Peano form of the remainder,is obviously a generalization of the definition of differentiability of a function at apoint, to which it reduces when n = 1.We remark that formula (5.79) is nearly always the more substantive of the two.For, on the one hand, as we have seen, it enables us to estimate the absolute magnitude of the remainder term.

On the other hand, when, for example, f (n+1) (x) isbounded in a neighborhood of x0 , it also implies the asymptotic formulaf (x0 )f (n) (x0 )(x − x0 ) + · · · +(x − x0 )n + O (x − x0 )n+1 .1!n!(5.81)Thus for infinitely differentiable functions, with which classical analysis deals in theoverwhelming majority of cases, formula (5.79) contains the local formula (5.80).In particular, on the basis of (5.81) and Examples 3–10 just studied, we can nowwrite the following table of asymptotic formulas as x → 0:f (x) = f (x0 ) +111x + x 2 + · · · + x n + O x n+1 ,1!2!n!11(−1)n 2nx + O x 2n+2 ,cos x = 1 − x 2 + x 4 − · · · +2!4!(2n)!11(−1)n 2n+1xsin x = x − x 3 + x 5 − · · · ++ O x 2n+3 ,3!5!(2n + 1)!1 2n1 2 1 4x + O x 2n+2 ,cosh x = 1 + x + x + · · · +2!4!(2n)!111x 2n+1 + O x 2n+3 ,sinh x = x + x 3 + x 5 + · · · +3!5!(2n + 1)!11(−1)n nx + O x n+1 ,ln(1 + x) = x − x 2 + x 3 − · · · +23nα(α−1)αα(α − 1) · · · (α − n + 1) nx2 + · · · +x(1 + x)α = 1 + x +1!2!n!+ O x n+1 .ex = 1 +Let us now consider a few more examples of the use of Taylor’s formula.Example 11 We shall write a polynomial that makes it possible to compute the values of sin x on the interval −1 ≤ x ≤ 1 with absolute error at most 10−3 .5.3 The Basic Theorems of Differential Calculus227One can take this polynomial to be a Taylor polynomial of suitable degree obtained from the expansion of sin x in a neighborhood of x0 = 0.

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