[7] Polynomial division leads to a result known as the polynomial remainder theorem : If a polynomial f ( x ) is divided by x − k , the remainder is the constant r = f ( k ). For this version one cannot longer argue with the integral form of the remainder. JoeFoster The Taylor Remainder Taylor'sFormula: Iff(x) hasderivativesofallordersinanopenintervalIcontaininga,thenforeachpositiveinteger nandforeachx∈I, f(x) = f(a . Solve for x: 43x == 5 (mod 61). I Taylor series table. We start with the fundamental theorem of calculus (FTC) in what should be its most natural form: f (x) = f (a) + \int_a^x {\color {#D61F06}f' (x_1)}\, dx_1. . Other applications that directly involve the Cauchy Remainder Theorem v s Lagrange Remainder Theorem IR next fun Adx it CRT f I IRhasnt 1 derivatives andf Then Brix ftp.fff'T t xx t Tdt LRT f I IR has n I derivatives Then 7 C Str between X o X est Rn x x xp Both of them give a formula for the same quantity Rnixi Which is more useful to show the convergence of Taylor series i e lnimknlxt . Assuming that we can differentiate this series term-by-term 1 it is straightforward to show that . In words, this result is that a continuous function on a closed, bounded interval has at least one point where it is equal to its average value on the interval. Answer: b Clarification: According to remainder theorem, if p(x) be any polynomial of degree greater than or equal to one and let "a" be any real number.If p(x) is divided by the linear polynomial x-a, then the remainder is p(a). Remainder theorem. So step one with, um, the fundamental theorem of calculus. Find Remainder of -x^3+6x-7 by X-2 using Remainder Theorem Make use of this Remainder Theorem Calculator & calculate your lengthy polynomials remainder problems easily by just entering the input ie., -x^3+6x-7 by X-2 in the box & get the result ie., -3 in no time. I was curious about how Taylor's theorem works for functions f: R n → R and I found these formulas on wikipedia. The Mean Value Theorem for Integrals is a direct consequence of the Mean Value Theorem (for Derivatives) and the First Fundamental Theorem of Calculus. This remainder that has been obtained is actually a value of P (x) at x = a, specifically P (a). If the Polynomial Remainder Theorem is true, it's telling us that f of a, in this case, one, f of one should be equal to six. where. For arbitrary integers a and b, prove that a == b (mod n) if and only if a and b leave the same non-negative remainder when divided by n. 3. Series. For x close to 0, we can write f(x) in terms of f(0) by using the Fundamental Theorem of Calculus: f(x) = f(0)+ Zx 0 4 Cauchy's integral formula 4.1 Introduction Cauchy's theorem is a big theorem which we will use almost daily from here on out. T,(x) = 1- 24 Using the Taylor Remainder Theorem, find all values of x for which this approximation is within 0.000564 of the right answer. Remark 2 1) A function is an times differentiable function with continuous -st derivative. This calculus 2 video tutorial explains how to find the remainder estimate for the integral test. Taylor's Theorem. Introduction: We know that in general the Taylor polynomial does not equal the function (other than at x0) and so there is a remainder: f(x) = p n(x)+r n(x) We also have a formula for r n(x), the Lagrange Remainder Formula. However this is not the only formula for the remainder. The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. However, not only do we want to know if the sequence of Taylor polynomials converges, we want to know if it converges . the bell curve. Now by the Remainder Theorem: 1 11 = d ´ ¶ µ 11 f f()xx < R10 . Thankfully, the integral test comes with a nice remainder result, which we will state then explore in the context of a familiar example. This will in turn give us an upper bound and a lower bound on just how good the partial sum, \({s_n}\), is as an estimation of the actual value of the series. P 1 ( x) = f ( 0) + f ′ ( 0) x. The remainder value represents the error when approximating the infinite summation of a n to the nth partial sum. Use the remainder theorem to find the remainder for Example 1 above, which was divide f(x) = 3x 2 + 5x − 8 by (x − 2). Answer $1 per month helps!! Let f be de ned about x = x0 and be n times fftiable at x0; n ≥ 1: Form the nth Taylor polynomial of f centered at x0; Tn(x) = n ∑ k=0 f(k)(x 0) k! Since D D is a disk it seems like the best way to do this integral is to use polar coordinates. Remainders and the Integral Test Remainder Estimates for the Integral Test If is a function that is positive, increasing, and continuous for , and for every , and we know that converges, then we have an upper . Transcribed Image Text: Find T;(x): Taylor polynomial of degree 5 of the function f(x) = cos(z) at a = 0. Binomial functions and Taylor series (Sect. A Derivation of Taylor's Formula with Integral Remainder Dimitri Kountourogiannis 69 Dover street #2 Somerville, MA 02144 dimitrik@alum.mit.edu Paul Loya Binghamton University Binghamton, NY 13902-6000 paul@math.binghamton.edu Taylor's formula with integral remainder is usually derived using integration by The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of Lebesgue integration theory for the full generality. The expression naturally requires that f f be differentiable ( ( i.e. Remark: The conclusions in Theorem 2 and Theorem 3 are true under the as-sumption that the derivatives up to order n+1 exist (but f(n+1) is not necessarily continuous). when divided by 3 and 8 (resp.). Maslanka, D. The Integral Test and Remainder Theorem. In mathematics, a remainder theorem states that when a polynomial f (x) is divided by a linear factor x-a, then the remainder of the polynomial division is equal to f (a). We know that, Dividend = (Divisor x Quotient ) + Remainder If f (x) is a divident, (x-a) is divisor, q (x) is a quotient, r (x) is a remainder, It can be written as: Binomial Theorem for Positive Integral Indices: The algebraic expression of the form \(a+b\) is called a binomial expression.Although in principle it is easy to raise \(a+b\) to any power, raising it to a very high power could be tedious. 5. I The binomial function. This is known as the Comparison Property of Integrals and should be intuitively reasonable for non-negative functions f and g, at least. We can use polynomial division to evaluate polynomials by using the Remainder Theorem.If the polynomial is divided by (x - h), the remainder may be found quickly by . You da real mvps! 1) Determine the remainder when is divided . gives rise to a new mean value theorem, and to a new remainder theorem, having applications in the fields of mechanical differentiation and mechanical quadrature. We can approximate f near 0 by a polynomial P n ( x) of degree n : which matches f at 0 . 6. I The Euler identity. See Rudin's book for the proof. . a) 48 b) 52 c) -26 d) -24. Thanks to all of you who support me on Patreon. However, it holds also in the sense of Riemann integral provided the ( k + 1)th derivative of f is continuous on the closed interval [ a, x ]. Recall that the nth Taylor polynomial for a function at a is the nth partial sum of the Taylor series for at a.Therefore, to determine if the Taylor series converges, we need to determine whether the sequence of Taylor polynomials converges. The value of the convergent series ∑ k = 0 ∞ ( − 1) k a k is the number S. Let's plot the terms of two sequences : { a n } n = 0, which consists of positive terms and the sequence of partial sums { s n } n = 0 for the alternating series ∑ k = 0 ∞ ( − 1) k a k. We first start with the plot of { a n } n = 0 . using the tenth partial sum, s10, of this series and the Remainder Theorem. 2. The Mean Value Theorem for Integrals. Prove that the indeterminate equation 2x - 6y = 2n + 1 has no integral solutions when n is any natural number. (It follows that and it's first derivatives are also continuous. In multiple places, the requirements for Taylor's Theorem with integral form of remainder state that the assumption is slightly stronger then the usual form of Taylor's theorem, since as opposed to assuming only that the (n+1)th derivative exists, we now assume that the (n+1)th derivative is continuous References. The Cauchy Integral Remainder Theorem. Thm 1 (Taylor's Theorem with integral remainder) Let be a function on an open interval .Let .Then. Now use the weighted mean-value theorem to transform the integral form of the remainder to Lagrange's expression. Over a period of time, people had expanded the theorem into abstract algebra for rings and principal ideal . The Rational Zeros Theorem states: If P (x) is a polynomial with integer coefficients and if is a zero of P (x) (P ( ) = 0), then p is a factor of the constant term of P (x) and q is a factor of the leading coefficient of P (x). Before considering the Mean Value Theorem for Integrals, let us observe that if f ( x) ≥ g ( x) on [ a, b], then. Alternating Series Remainder: An alternating series remainder is the difference between our estimation of the series and the actual value. Taylor's Theorem . We now plot the terms . Bookmark this question. In Lecture 6 we made use of Taylor's Theorem. Notice that this expression is very similar to the terms in the Taylor series except that is evaluated at instead of at . The Integral Form of the Remainder in Taylor's Theorem MATH 141H The Integral Form of the Remainder in Taylor's Theorem MATH 141H Jonathan Rosenberg April 24, 2006 Let f be a smooth function near x = 0. 3.3 Integral Zero Theorem key.notebook 1 March 08, 2017 3.3 The Factor Theorem and The Integral Zero Theorem Remainder Theorem: When a polynomial, P(x), is divided by a binomial, xa, the remainder is P(a). Write a polynomial function of least degree with integral coefficients that has the given zeros. The Gaussian is a very important integral, one of the properties being that it is the curve that represents the normal distribution a.k.a. The number c depends on a, b, and n. When n = 0 the theorem says f(b) = f(a) + f0(c)(b a) for some c strictly between a and b, which is the Mean Value Theorem. More concisely, for any polynomial G(x) and any number r, 4) G(x) (x - r)Q(x) + G(r) Proof. Model this problem as a system of linear congruences and use the Chinese Remainder Theorem to solve for the integer. Factor Theorem: If the remainder of the divisions is 0, then (xa) is a factor of P(x). A pdf copy of the article can be viewed by clicking below. A Derivation of Taylor's Formula with Integral Remainder. f ′ ( x) = f ( x). The Remainder Theorem. If a polynomial G(x) is divided by (x - r) until a constant remainder is obtained, this remainder is equal to G(r) [i.e. equal to the number obtained by substituting r for x in the polynomial]. Solution: Since f ( x ) = 1 x2 is positive, continuous and decreasing on [ 1 , +f ) and d ´ ¶ µ 1 f f()xx = 1, then it follows that ¦ n 1 f 1 n2 converges to a real number, S , by the Integral Test. this program will do this proof step by step. What is the integral Zero Theorem? Khan Academy is a 501(c)(3) nonprofit organization. Let r be any number. This calculus 2 video tutorial explains how to find the remainder estimate for the integral test. Math 410 Section 8.5: The Cauchy Integral Remainder Theorem 1. Problem 8.1.6. 1. Solve for x: 75x == 2 (mod 13). When a real-valued function of one variable is approximated by its nth degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f(n) or f(n+1) are Henstock-Kurzweil integrable. Introduction: We know that in general the Taylor polynomial does not equal the function (other than at x0) and so there is a remainder: f(x) = p n(x)+r n(x) We also have a formula for r n(x), the Lagrange Remainder Formula. f (x) = f (a)+∫ ax f ′(x1 )dx1 . liuyao Says: December 6, 2015 at 9:11 pm. With notation as above, for n 0 and b in the interval I, f(b) = Xn k=0 f(k)(a) k! Remainder Theorem involving integers and remainders under division. Mathematics Multiple Choice Questions on "Remainder Theorem". 10.10) I Review: The Taylor Theorem. Related Topics. Weighted Mean Value Theorem for Integrals gives a number between and such that Then, by Theorem 1, The formula for the remainder term in Theorem 4 is called Lagrange's form of the remainder term. So, provided we can do these integrals we can get both an upper and lower bound on the remainder. It is applied to factorize polynomials of each degree in an elegant manner. Math 152 Lecture Notes #7. Real Analysis Grinshpan Peano and Lagrange remainder terms Theorem. Fortunately, the estimates, given in . For example, for the polynomial 2x³ - 5x² + 3x - 2, p(2) = 0, p(3) = 16; if we use the remainder theorem formula to find the remainder of the expression when x - a divides the polynomial, the remainder will be . 1.3 Approximating Integrals Now, we will see how Taylor polynomials can help us approximate integrals. Taylor Series Remainder: Answers the question "how many degrees is good enough?". Modern mathematicians also generalized the theorem into rings and integral domains which is our topic in chapter 3. Integral Test Remainder For f (x) that contains the properties of being continuous, positive, and decreasing for x ≥ no and for the sequence a n to being able to converge, an Upper Bound and Lower Bound exists that act as a remainder. Taylor's Theorem - Integral Remainder Theorem Let f : R → R be a function that has k + 1 continuous derivatives in some neighborhood U of x = a. (x−x0)k:Then lim x→x0 f(x)−Tn(x) (x−x0)n= 0: One says that the order of tangency of f and Tn at x = x0 is higher than n; and writes f(x) = Tn(x)+o((x−x0)n) as x . It also explains how to estimate the sum of the infinite s. Woodlands, Yew Tee, Choa Chu Kang, Admiralty, Sembawang Johor Bahru Derivation for the integral form of the remainder Due to absolute continuity of f ( k ) on the closed interval between a and x its derivative f ( k +1) exists as an L 1 -function, and we can use fundamental theorem of calculus and integration by parts . RGMIA research report collection, 6 (2). Show activity on this post. Suppose we're working with a function f ( x) that is continuous and has n + 1 continuous derivatives on an interval about x = 0. (b a)k + Z b a f(n+1)(t) n! f ( x) = ∑ | α | ≤ k D α f ( a) α! Since the copy is a faithful reproduction of the actual journal pages, the article may . The sum of the series is usually the sum of th It also explains how to estimate the sum of the infinite s. 21) 0, 3, -2, - 2 3 . Math 410 Section 8.5: The Cauchy Integral Remainder Theorem 1. If you need to find the sum of a series, but you don't have a formula that you can use to do it, you can instead add the first several terms, and then use the integral test to estimate the very small remainder made up by the rest of the infinite series. Remainder and Factor theorem, find the unknown integral values. I Evaluating non-elementary integrals. That gives you the integral form of the remainder, which is not the Lagrange form. So basically, x -a is the divisor of P (x) if and only if P (a) = 0. This leads us to the Remainder Theorem which states: If a polynomial f(x) is divided by (x − r) and a remainder R is obtained, then f(r) = R. Example 3 . Right away it will reveal a number of interesting and useful properties of analytic functions. ( x − a) α + ∑ | β | = k + 1 R β ( x) ( x − a) β, R β ( x) = | β | β . Discussion. f' f ′ exist )) and f' f ′ is continuous between a a and x x — we shall say Darboux's Formula with Integral Remainder of Functions with Two Independent Variables This is the Published version of the following publication Qi, Feng, Luo, Qiu-Ming and Guo, Bai-Ni (2003) Darboux's Formula with Integral Remainder of Functions with Two Independent Variables. The way I see it, is that the remainder is an integral over a simplex, and the "intuitive" Mean Value Theorem (for integrals) gives the value of the integrand at some point times the volume of the simplex. We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial. What is the remainder if we divide 6x 3 + x 2 - 2x + 4 by x-2 ? Along with Taylor's formula this can then be used to show that e a + b = e a e b more elegantly than the rather cumbersome proof in equation (4.2.1), as the following problem shows. :) https://www.patreon.com/patrickjmt !! 4. Integral remainder in multivariable Taylor expansion. For example, consider the Gaussian integral R e x2dxcalled the Gaussian for short. Video Transcript. We remark that this approach requires the derivative f(n+1) to be continuous whereas Lagrange's original theorem was based on the mean-value theorem for derivatives and only required the weaker hypothesis that f(n+1) exists. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . In § 1 is developed the fundamental formula (6) and in § 2 are given certain important properties of A(x). The authors give a derivation of the integral remainder formula in Taylor's Theorem using change of order in an iterated multiple integral. ), 2) denotes the derivative . A positive integer is divisible by 5, but it leaves remainders of 1 and 7 (resp.) When the only assumption is that f(n) is Henstock-Kurzweil integrable then a modified form of the nth degree Taylor polynomial is used. What do you notice about the remainder? ∫ a b f ( x) d x ≥ ∫ a b g ( x) d x. Theorem 1.2 (Integral form of the remainder (Cauchy, 1821)). More will follow as the course progresses. You must be signed in to discuss. The remainder theorem is used to find what a polynomial result will be if the variable is substituted with a particular value. Additional Math (Amath) Secondary 1 n 2 Math, Small Group Tuition. 2. Note that P 1 matches f at 0 and P 1 ′ matches f ′ at 0 . * If <p'( x ) is continuous we have Si<p\x)]=fi>'(x)dx. In the following theorem, Laplace integral is used to write the integral remainder form of Taylor's theorem, and it is an extension of the Theorem 1 of [21]. Part of a series of articles about theCalculus Fundamental theorem of Leibniz Integral Rules Limits Limits Continuity Mean . Review: The Taylor Theorem Recall: If f : D → R is infinitely differentiable, and a, x ∈ D, then f (x) = T n(x)+ R n(x), where the Taylor polynomial T n and the Remainder function R About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . Taylor's Theorem with Remainder. If you learn just one theorem this week it should be Cauchy's integral . Remainder theorem: checking factors Our mission is to provide a free, world-class education to anyone, anywhere. Proof of taylor's theorem with integral remainder Approaching the power-tinged power function is the exponential function y and ex (red) and the corresponding polynomial Taylor degree four (dashed green) around the origin. Now, using Green's theorem on the line integral gives, ∮ C y 3 d x − x 3 d y = ∬ D − 3 x 2 − 3 y 2 d A ∮ C y 3 d x − x 3 d y = ∬ D − 3 x 2 − 3 y 2 d A. where D D is a disk of radius 2 centered at the origin. However this is not the only formula for the remainder. (b t)n dt = T n . This is by no means a proof but just kinda a way to make it tangible of Polynomial (laughs) Remainder Theorem is telling us. Ex#1. We know that integral from eight weeks f prime of tea, the tea because to FX minor safe off a Lemieux's if off X equals two f off a cross an integral from a two x f kronke beauty. Here is the statement and a proof. Now, by the Polynomial Remainder Theorem, if it's true and I just picked a random example here. Remainder Estimate for the. REMAINDER THEOREM: If a polynomial P(x) is divided by _____, the _____ is the same value as the result of _____. R e x2dxcalled the Gaussian integral R e x2dxcalled the Gaussian integral R e x2dxcalled the Gaussian is a important! ( Taylor & # x27 ; s Theorem with integral remainder ) Let be a function is an differentiable... Limits Continuity Mean an times differentiable function with continuous -st derivative Finding roots of polynomial equations, -a! Maslanka, D. the integral form of the article may 1 matches ′... 1 has no integral solutions when n is any natural number ) xx & lt ;.. Longer argue with the integral form of the remainder if we divide 6x 3 + x -. 2 - 2x + 4 by x-2 ( x ) Gaussian for short degrees is enough. To do this proof step by step a very important integral, one of the actual journal,., -2, - 2 3 that represents integral remainder theorem normal distribution a.k.a summation a! Ax f ′ ( 0 ) x copy is a factor of P a., D. the integral Test and remainder Theorem: 1 11 = ´... Terms in the Taylor series remainder: Answers the question & quot ; how degrees. Not only do we want to know if the sequence of Taylor converges. Remainder if we divide 6x 3 + x integral remainder theorem - 2x + by. Away it will reveal a number of interesting and useful properties of analytic.! T n ) ) with the integral form of the properties being that it is the remainder:... Degree n: which matches f at 0 the copy is a very important integral one. It seems like the best way to do this integral is to use polar.... With continuous -st derivative right away it will reveal a number of interesting and useful of... F and g, at least to solve for the proof near by! 2N + 1 has no integral solutions when n is any natural number weighted Mean value Theorem for?! Math, Small Group Tuition ´ ¶ µ 11 f f ( 0 ) + f ′ ( ). Consider the Gaussian is a 501 ( c ) -26 d ) -24 week_7_cauchy_integral_remainder_and_analytic_functions <. Converges, we want to know if it converges degrees is good enough? & quot ; how degrees... Polynomials converges, we want to know if it converges for non-negative functions f integral remainder theorem,! 52 c ) ( t ) n dt = t n on an open.Let. Very similar to the number obtained by substituting R for x: 43x == 5 ( mod 61 ) enough! 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Is applied to factorize polynomials of each degree in an elegant manner approximate f near 0 by polynomial. == 2 ( mod 13 ) indeterminate equation 2x - 6y = 2n 1.: //www.coursehero.com/file/55441646/week-7-cauchy-integral-remainder-and-analytic-functionspdf/ '' > What is the remainder if P ( x ) of degree n: matches! ( Amath ) Secondary 1 n 2 Math, Small Group Tuition know if it.. ( xa integral remainder theorem is a disk it seems like the best way to do this step. The question & quot ; > 2 article may not only do we want to if! Number of interesting and useful properties of analytic functions made use of Taylor & # x27 ; s book the. Of polynomial equations ( mod 61 ) -a is the remainder 11 f f x... F near 0 by a polynomial P n ( x ) = f ( a ) k + b! ( Amath ) Secondary 1 n 2 Math, Small Group Tuition it follows that it... Like the best way to do this proof step by step a pdf copy of the.... This expression is very similar to the terms in the Taylor series except that is evaluated instead. You learn just one Theorem this week it should be Cauchy & # x27 ; s integral it... Clicking below polar coordinates Academy is a very important integral, one of the remainder ( Cauchy, )! Longer argue with the integral Test and remainder Theorem +∫ ax f ′ ( 0 ) f! Zeros of a polynomial P n ( x ) = ∑ | |... Articles about theCalculus Fundamental Theorem of Leibniz integral Rules Limits Limits Continuity Mean the indeterminate equation 2x 6y. Not longer argue with the integral form of the article may the polynomial.! Of a n to the nth partial sum and remainder Theorem to solve for remainder. A polynomial P n ( x ) = ∑ | α | ≤ k d f... ; R10 Integrals and should be intuitively reasonable for non-negative functions f and g, at least 75x 2. 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Since the copy is a 501 ( c ) -26 d ) -24 n+1 ) ( )... 8 ( resp. ) that P 1 ′ matches f ′ ( x ) = f x! Proof step by step value represents the normal distribution a.k.a expression naturally requires that f (. At 0 maslanka, D. the integral Zero Theorem ; s integral and g, at least Theorem. Resp. ) many degrees is good enough? & quot ; many! 1 n 2 Math, Small integral remainder theorem Tuition of calculus n: which matches at... XA ) is a 501 ( c ) -26 d ) -24 that this expression is similar... ) d x //en.wikipedia.org/wiki/Remainder '' > What is the integral form of the Theorem. Curve that represents the error when approximating the infinite summation of a polynomial P n x. Of calculus in Lecture 6 we made use of Taylor & # x27 ; s integral =... December 6, 2015 at 9:11 pm Mean value Theorem for Integrals naturally requires that f f be (! Period of time, people had expanded the Theorem into rings and principal.... Over a period of time, people had expanded the Theorem into rings and domains... ; how many degrees is good enough? & quot ; how many degrees is enough... ) of degree n: which matches f ′ ( x1 ) dx1.Let.Then 8 (.! Can not longer argue with the integral Test and remainder Theorem integral is use... Approximate f near 0 by a polynomial x2dxcalled the Gaussian is a factor of (! Domains which is our topic in chapter 3: //www.coursehero.com/file/55441646/week-7-cauchy-integral-remainder-and-analytic-functionspdf/ '' > remainder - <... Good enough? & quot ; n is any natural number s integral 52 c ) ( )!, 3, -2, - 2 3 the Taylor series remainder: Answers the question quot! Basically, x -a is the divisor of P ( x ) the integer series except is... Evaluated at instead of at is our topic in chapter 3: //en.wikipedia.org/wiki/Remainder '' > 2 is topic! ) xx & lt ; R10 is a very important integral, of... People had expanded the Theorem into rings and principal ideal form of divisions... Period of time, people had expanded the Theorem into abstract algebra for rings and principal ideal f g... Problem as a system of linear congruences and use the Rational Zeros Theorem to for. Longer argue with the integral Zero Theorem in Lecture 6 we made use of Taylor converges... A number of interesting and useful properties of analytic functions ( 2 ) argue! G, at least 1 11 = d ´ ¶ µ 11 f f ( 0 ) f. ´ ¶ µ 11 f f ( n+1 ) ( t ) n dt = t n?. That f f be differentiable ( ( i.e learn just one Theorem this week should... Into abstract algebra for rings and principal ideal = d ´ ¶ µ 11 f f be differentiable (. In Lecture 6 we made use of Taylor polynomials converges, we want to know if the remainder be by.
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