2024 Taylor series expansion

That's the coefficient on this. Plus-- this is the power rule right here-- 2 times 1/2 is just 1, plus f prime prime of 0 times x. Take the 2, multiply it times 1/2, and decrement that 2 right there. I think you now have a sense of why we put the 1/2 there. . Tommy prine

Taylor Series for Common Functions. ... We call these Taylor series expansions, or Taylor series. We will compute the Taylor series of several functions, all centered at 0. Recall from our previous practice the following steps: First, compute the ...dy(t) =. dt. f (t, y(t)) (1.1) y(0) = y0. This equation can be nonlinear, or even a system of nonlinear equations (in which case y is. a vector and f is a vector of n different functions). Numerical Solution of an ODE: The idea behind numerical solutions of a Differential Equation is to replace differentiation by differencing.The Taylor Series in (x − a) ( x − a) is the unique power series in (x − a) ( x − a) converging to f(x) f ( x) on an interval containing a a. For this reason, By Example 1, e−2x = 1 − 2x + 2x2 − 4 3x3 + … e − 2 x = 1 − 2 x + 2 x 2 − 4 3 x 3 + …. where we have substituted −2x − 2 x for x x . By Example 2, since d dx ... Dec 29, 2020 · Since the Taylor series for $\sin x$ has an infinite radius of convergence, so does the Taylor series for $\sin(x^2)$. The Taylor expansion for $\ln x$ given in Key Idea 32 is centered at $x=1$,so we will center the series for $\ln (\sqrt{x})$ at $x=1$ as well. May 3, 2023 · Theorem $\PageIndex{1}$: Taylor’s Theorem (Taylor Series) The uniqueness of Taylor series along with the fact that they converge on any disk around $z_0$ where the function is analytic allows us to use lots of computational tricks to find the series and be sure that it converges. Taylor & Francis is a renowned publisher in the academic and research community, offering an extensive collection of journals covering a wide range of disciplines. Taylor & Francis...When the Eras tour rolled around (speculatively Taylor Swift's last concert tour ever), I booked a spontaneous trip to Phoenix, Arizona. We may be compensated when you click on pro...طريقة إيجاد صورة الرقم من خلال استخدام Taylor Series.Khaled Al Najjar , Pen&Paperلاستفساراتكم واقتراحاتكم :Email: khaled.civil95@gmail ...Taylor series are extremely powerful tools for approximating functions that can be difficult to compute otherwise, as well as evaluating infinite sums and integrals by recognizing Taylor series. If only concerned about the neighborhood very close to the origin, the $n=2$ approximation represents the sine wave sufficiently, and no higher ...Taylor Series Expansions In this short note, a list of well-known Taylor series expansions is provided. We focus on Taylor series about the point x = 0, the so-called Maclaurin series. In all cases, the interval of convergence is indicated. The variable x is real. We begin with the inﬁnite geometric series: 1 1− x = X∞ n=0 xn, |x| < 1. (1)As concern for the environment and sustainability continues to grow, more and more fashion companies are making efforts to reduce their impact on the planet. One of these companies...taylor expansion of arctan (x) Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music….The above Taylor series expansion is given for a real values function f(x) where f’(a), f’’(a), f’’’(a), etc., denotes the derivative of the function at point a. If the value of point ‘a’ is zero, then the Taylor series is also called the Maclaurin series. Get ratings and reviews for the top 11 pest companies in Taylor, MI. Helping you find the best pest companies for the job. Expert Advice On Improving Your Home All Projects Feature...Taylor & Francis is a renowned academic publisher that has been providing researchers, scholars, and professionals with access to high-quality scholarly journals for over two centu...The Taylor series can sometimes be called a MacLaurin series, which is a Taylor series evaluated at a = 0. In this lesson, we have used the Taylor series to approximate a function and to ...This calculus 2 video tutorial explains how to find the Taylor series and the Maclaurin series of a function using a simple formula. It explains how to deri...power-series; taylor-expansion; elementary-functions; Share. Cite. edited Feb 2, 2017 at 3:59. smaude. asked Feb 2, 2017 at 3:38. smaude smaude. 869 3 3 gold badges 10 10 silver badges 16 16 bronze badges $\endgroup$ 9. 5 $\begingroup$ A Taylor series exists if and only if the function is infinitely differentiable at some a. $\endgroup$Arguments. character, or function returning a numeric scalar value. vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated (the center of the Taylor series). See derivative. list of additional parameters passed to f. the order of the Taylor approximation.Jeremy Orloﬀ. 7 Taylor and Laurent series. 7.1 Introduction. We originally deﬁned an analytic function as one where the derivative, deﬁned as a limit of ratios, existed. We went on to prove Cauchy’s theorem and Cauchy’s integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of ...A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) …Assuming "Taylor series" is a general topic | Use as a computation or a calculus result or referring to a mathematical definition instead. Examples for Series Expansions. Taylor Series. Find a Taylor series expansion: taylor series sin x. Expand around a specified point: series sin x at x=pi/4. Specify the order of the expansion:Wolfram|Alpha Widgets: "Taylor Series Calculator" - Free Mathematics Widget. Taylor Series Calculator. Added Nov 4, 2011 by sceadwe in Mathematics. A calculator for finding the expansion and form of the Taylor Series of a given function. To find the Maclaurin Series simply set your Point to zero (0). In the ever-evolving world of academia, access to high-quality research materials is paramount. Scholars and researchers rely on reputable journals to stay updated on the latest ad...When it comes to finding the perfect pair of men’s jeans, quality and craftsmanship are essential. David Taylor has been a trusted name in men’s fashion for years, known for their ...Free online series calculator allows you to find power series expansions of functions, providing information you need to understand Taylor series, Laurent series, Puiseux series and more. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.Medicine Matters Sharing successes, challenges and daily happenings in the Department of Medicine Casey Overby Taylor, Ph.D., assistant professor in the Division of General Interna...Sep 5, 2021 · Taylor's Theorem thus states: f(x) = n − 1 ∑ k = 0f ( k) (a)(x − a)k k! + f ( n) (c)(x − a)n n! The important point to realize here is that n stands for an integer, such that a finite differentiable function can be expressed as a series of its n derivatives evaluated at some point a. 1 Derivation of Taylor Series Expansion Objective: Given f(x), we want a power series expansion of this function with respect to a chosen point xo, as follows: (1) (Translation: find the values of a0, a1, a2, … of this infinite series so that the equation holds. Method: The general idea will be to process both sides of this equation and choose values of x so that …In the world of fashion, finding a brand that combines both style and quality can be a challenging task. However, if you are someone who appreciates timeless designs and impeccable...Arguments. character, or function returning a numeric scalar value. vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated (the center of the Taylor series). See derivative. list of additional parameters passed to f. the order of the Taylor approximation.The Taylor Series in (x − a) ( x − a) is the unique power series in (x − a) ( x − a) converging to f(x) f ( x) on an interval containing a a. For this reason, By Example 1, e−2x = 1 − 2x + 2x2 − 4 3x3 + … e − 2 x = 1 − 2 x + 2 x 2 − 4 3 x 3 + …. where we have substituted −2x − 2 x for x x . By Example 2, since d dx ... 24 Jul 2015 ... The "a" is the number where the series is "centered". There are usually infinitely many different choices that can be made for a, though the&nbs...so the binomial series for 1 √1 + x is given by. 1 √1 + x = 2 d dx√1 + x = 1 + ∞ ∑ n = 1( − 1)n n! 1 ⋅ 3 ⋅ 5⋯(2n − 1) 2n xn. Exercise 7.5.3. Find the binomial series for f(x) = 1 (1 + x)3 / 2. Hint. Answer. In this example, we differentiated a known Taylor series to construct a Taylor series for another function.Revising derivation in Taylor's series expansion.Go to http://www.examsolutions.net/ for the index, playlists and more maths videos on Taylors series...First moment. Given and , the mean and the variance of , respectively, [1] a Taylor expansion of the expected value of can be found via. Since the second term vanishes. Also, is . Therefore, . It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example, Learn how Taylor polynomials can help you understand the essence of calculus and the behavior of functions. This video explains the intuition and the formula behind Taylor series, with clear ... The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is ...Taylor Series Expansion. Taylor models are estimators made by the sum of a multivariate polynomial part, that matches the Taylor series expansion of the estimated function up to a specified order (typically around the interval midpoint), and a remainder interval, which provides a bound on the higher-order terms [11].Definition 10.3.1: Maclaurin and Taylor series. If f has derivatives of all orders at x = a, then the Taylor series for the function f at a is. ∞ ∑ n = 0f ( n) (a) n! (x − a)n = f(a) + f′ (a)(x − a) + f ″ (a) 2! (x − a)2 + ⋯ + f ( n) (a) n! (x − a)n + ⋯. The Taylor series for f at 0 is known as the Maclaurin series for f.taylor series expansion of e^x. Natural Language; Math Input; Extended Keyboard Examples Upload Random. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…Taylor Expansion. If a function has continuous derivatives up to (n+1)th order, then this function can be expanded in the following fashion: where , called the remainder after n+1 terms, is given by: When this expansion converges over a certain range of , that is, , then the expansion is called the Taylor Series of expanded about .f ″ (0) = 0. etc. So be applying Taylor's theorem here, one has. e − 1 / x2 = 0. which is nonsense. Then, there is a second case. As Ethan Alwaise mentions, any series expansion makes no sense if it doesn't converge. Take, for example, the expansion of 1 1 − r at r = 0. Then consider that expansion for r = 2.Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history ... Tyler Perry ‘s planned $800 million studio expansion in Atlanta has been put on the back burner due to his growing concerns over developments in artificial …series is 1. Instead of deriving this from the formula for the geometric series we could also have computed it using Taylor’s formula. Try it! Question: If you put in −1 for x the series diverges. If you put in 1, it looks like it would converge. Answer: The graph of y = 1+ 1 x looks smooth at x = 1, but there is still a problem.A Taylor series is a way of expressing a function as a power series using its derivatives. Recall that a power series is a sum of the form {eq}\sum_ {n=0}^ {\infty} c_n (x-a)^n {/eq}. A Taylor ...The Taylor expansion or Taylor series representation of a function, then, is. sin (x) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. Here, is the factorial of and denotes the th derivative of at the point . If this series converges for every in the interval and the sum is equal to , then the function is called analytic.Sep 5, 2021 · The special type of series known as Taylor series, allow us to express any mathematical function, real or complex, in terms of its n derivatives. The Taylor series can also be called a power series as each term is a power of $x$, multiplied by a different constant \[ f(x) = a_0x^0 +a_1x^1 +a_2x^2 +a_3x^3 +... a_nx^n \] Jeremy Orloﬀ. 7 Taylor and Laurent series. 7.1 Introduction. We originally deﬁned an analytic function as one where the derivative, deﬁned as a limit of ratios, existed. We went on to prove Cauchy’s theorem and Cauchy’s integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of ... Yellowstone, the hit TV series created by Taylor Sheridan, has taken the world by storm. With its captivating storyline and intriguing characters, it has become a favorite among au...Taylor Series. A Taylor Series is an expansion of some function into an infinite sum of terms, where each term has a larger exponent like x, x 2, x 3, etc. Example: The Taylor Series for ex. ex = 1 + x + x22! + x33! + x44! + x55! + ... says that the function:ex.Sep 5, 2021 · The special type of series known as Taylor series, allow us to express any mathematical function, real or complex, in terms of its n derivatives. The Taylor series can also be called a power series as each term is a power of $x$, multiplied by a different constant \[ f(x) = a_0x^0 +a_1x^1 +a_2x^2 +a_3x^3 +... a_nx^n \] Taylor Series Expansions In this short note, a list of well-known Taylor series expansions is provided. We focus on Taylor series about the point x = 0, the so-called Maclaurin series. In all cases, the interval of convergence is indicated. The variable x is real. We begin with the inﬁnite geometric series: 1 1− x = X∞ n=0 xn, |x| < 1. (1) so the binomial series for 1 √1 + x is given by. 1 √1 + x = 2 d dx√1 + x = 1 + ∞ ∑ n = 1( − 1)n n! 1 ⋅ 3 ⋅ 5⋯(2n − 1) 2n xn. Exercise 7.5.3. Find the binomial series for f(x) = 1 (1 + x)3 / 2. Hint. Answer. In this example, we differentiated a known Taylor series to construct a Taylor series for another function.Then if the function $f$ has $n+1$ derivatives on an interval that contains both $x_0$ and $x\text{,}$ we have the Taylor expansion ... When $x_0=0$ this is also called the Maclaurin series for $f\text{.}$ Here are Taylor series …Taylor Expansion. If a function has continuous derivatives up to (n+1)th order, then this function can be expanded in the following fashion: where , called the remainder after n+1 terms, is given by: When this expansion converges over a certain range of , that is, , then the expansion is called the Taylor Series of expanded about . Not only does Taylor’s theorem allow us to prove that a Taylor series converges to a function, but it also allows us to estimate the accuracy of Taylor polynomials in approximating function values. We begin by looking at linear and quadratic approximations of f ( x ) = x 3 f ( x ) = x 3 at x = 8 x = 8 and determine how accurate these ... Medicine Matters Sharing successes, challenges and daily happenings in the Department of Medicine Nadia Hansel, MD, MPH, is the interim director of the Department of Medicine in th...power-series; taylor-expansion; Share. Cite. Follow edited Oct 29, 2015 at 18:57. Educ. asked Oct 29, 2015 at 10:07. Educ Educ. 4,770 4 4 gold badges 25 25 silver badges 42 42 bronze badges $\endgroup$ Add a comment | 2 Answers Sorted by: Reset to default 8 $\begingroup$ How about a rather simple derivation like the one below: ...Matthew Hochberg. 11 years ago. No, you just know the Taylor series at a specific point (also the Maclaurin series) or, to be more clear, each succeeding polynomial in the series will hug more and more of the function with the specified point that x equals being the one point that every single function touches (in the video above, x equals 0).Assuming "taylor series" is a general topic ... Find a Taylor series expansion: taylor series sin x. Expand around a specified point: series sin x at x=pi/4. In the world of academic publishing, Taylor & Francis stands out as a leading publisher that offers numerous advantages for both researchers and authors. One of the primary advanta...Yellowstone, the critically acclaimed television series created by Taylor Sheridan, has captured the hearts of millions of viewers with its gripping storyline and breathtaking cine...18.4.1 Summary. 1. Some functions can be perfectly represented by a Taylor series, which is an infinite sum of polynomials. 2. Functions that have a Taylor series expansion can be approximated by truncating its Taylor series. 3. The linear approximation is a common local approximation for functions. 4.Taylor’s Theorem and Taylor’s Series is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.Yellowstone, the critically acclaimed television series created by Taylor Sheridan, has captured the hearts of millions of viewers with its gripping storyline and breathtaking cine...A Taylor expansion of a function f f around some value x0 x 0 is similar to a prediction of the function at a neighboring value x x knowing progressively more about the variation of f f at the point x0 x 0. First step: easiest prediction: nothing changed, that is, f(x) = f(x0) f ( x) = f ( x 0)A First Look at Taylor Series. Most ``smooth'' functions can be expanded in the form of a Taylor series expansion : This can be written more compactly as ' is pronounced `` is given in Appendix E signals are in that category, because hearing is bandlimited kHz, and the audible spectrum of any sum of sinusoids is infinitely differentiable.Learn what is Taylor series expansion of a function, how to use it to approximate the values of a function around different points, and how it is related to calculus and machine learning. See …Examples of Taylor series. Taylor Series. David Sinden · January 11, 2024 · 10 minute read . Taylor Series Expansion. This notebook presents the Taylor series expansion of the sine function, $\sin\left( x\right)$ close to zero for an increasing number of terms in the approximation.1. Expand the function f(z) = 2(z + 2) z2 − 4z + 3 in a Taylor series about the point z = 2 and find the circle C inside of which the series converges. Find a Laurent series that converges in the region outside of C. I tried writing the denominator as (z − 3)(z − 1) to find the singularities z = 1, 3 both simple.This calculus 2 video tutorial explains how to find the Taylor series and the Maclaurin series of a function using a simple formula. It explains how to deri...This is when series expansions such as the Taylor series come in handy – these tricky functions can now be written as the sum of infinite polynomials! The Taylor series is an infinite series that can be used to rewrite transcendental functions as a series with terms containing the powers of $\boldsymbol{x}$. In fact, through the Taylor series ... Link Taylor was one of four divested brand names under the Dixie Furniture Company, and the Link Taylor brand division manufactured fine solid wood furnishings for the dining room ...Dec 10, 2016 · 10. The Taylor expansion is one of the most beautiful ideas in mathematics. The intuition is simple: most functions are smooth over ranges we’re interested in. And polynomials are also smooth ... 📒⏩Comment Below If This Video Helped You 💯Like 👍 & Share With Your Classmates - ALL THE BEST 🔥Do Visit My Second Channel - https://bit.ly/3rMGcSAThis vi... 3.6.1 Taylor expansion. A Taylor series is a series of polynomial terms to approximate the function ( f ( x )) at a point ( x = a ). A one-dimensional Taylor series is given by. (3.25) If a is set as 0, the corresponding Taylor expansion is known as a Maclaurin series, (3.26)Taylor series expansion at Infinity. 2. Series expansion of $\ln(1+(1-x)^{1/2})$ 0. Taylor Series expansion of $\frac{(1+x)}{(x-1)^2}$ Hot Network Questions Is it true that no philosopher disagrees that everything exists? What's the phrase "10 years" used for in the sentence "I'm 10 years your senior" What theorems from single-variable …Jul 13, 2020 · Definition 5.4.1: Maclaurin and Taylor series. If f has derivatives of all orders at x = a, then the Taylor series for the function f at a is. ∞ ∑ n = 0f ( n) (a) n! (x − a)n = f(a) + f′ (a)(x − a) + f ″ (a) 2! (x − a)2 + ⋯ + f ( n) (a) n! (x − a)n + ⋯. The Taylor series for f at 0 is known as the Maclaurin series for f.

Series Expansions. Wolfram|Alpha can compute Taylor, Maclaurin, Laurent, Puiseux and other series expansions. A series expansion is a representation of a mathematical expression in terms of one of the variables, often using the derivative of the expression to compute successive terms in the series. A partial sum of a series expansion can be ... . Send current location iphone

The limitations of Taylor's series include poor convergence for some functions, accuracy dependent on number of terms and proximity to expansion point, limited radius of convergence, inaccurate representation for non-linear and complex functions, and potential loss of efficiency with increasing terms. Expansion around a point, and some common Taylor series. A common situation for us in applying this to physics problems will be that we know the full solution for some system in a simplified case, and then we want to turn on a small new parameter and see what happens. We can think of this as using Taylor series to approximate $ f(x_0 + \epsilon) $ when …Wolfram|Alpha Widgets: "Taylor Series Calculator" - Free Mathematics Widget. Taylor Series Calculator. Added Nov 4, 2011 by sceadwe in Mathematics. A calculator for finding the expansion and form of the Taylor Series of a given function. To find the Maclaurin Series simply set your Point to zero (0).The original function f(x) f ( x) is approximated by f~(x) f ~ ( x). The approximation is accurate near the expansion point x0 x 0. Figure 12.2 12. 2: Taylor series expansions of the function f(x) = 1/(1 − x) f ( x) = 1 / ( 1 − x) about x x = 0. Accuracy increases as more terms are retained. The thick curve is the exact function.📒⏩Comment Below If This Video Helped You 💯Like 👍 & Share With Your Classmates - ALL THE BEST 🔥Do Visit My Second Channel - https://bit.ly/3rMGcSAThis vi...Learn how Taylor polynomials can help you understand the essence of calculus and the behavior of functions. This video explains the intuition and the formula behind Taylor series, with clear ... Definition 10.3.1: Maclaurin and Taylor series. If f has derivatives of all orders at x = a, then the Taylor series for the function f at a is. ∞ ∑ n = 0f ( n) (a) n! (x − a)n = f(a) + f′ (a)(x − a) + f ″ (a) 2! (x − a)2 + ⋯ + f ( n) (a) n! (x − a)n + ⋯. The Taylor series for f at 0 is known as the Maclaurin series for f.Taylor Series Expansion Calculator computes a Taylor series for a function at a point up to a given power. Taylor series is a way to represent a function as a sum of terms calculated based on the function's derivative values at a given point as shown on the image below. Taylor series calculator present the computed Taylor series as sum of its ...May 3, 2023 · Theorem $\PageIndex{1}$: Taylor’s Theorem (Taylor Series) The uniqueness of Taylor series along with the fact that they converge on any disk around $z_0$ where the function is analytic allows us to use lots of computational tricks to find the series and be sure that it converges. 1 day ago · A one-dimensional Taylor series is an expansion of a real function f (x) about a point x=a is given by (1) If a=0, the expansion is known as a Maclaurin series. Taylor's theorem (actually discovered first by Gregory) states that any function satisfying certain conditions can be expressed as a Taylor series. Taylor Series Expansion. In financial markets participants would like to measure the effect of changes in the price of the bond due to changes in yield. This enables better risk management of financial assets as the impact of asset values is determinable. Recomputing the value of the bond using the changed yield comes across as an obvious solution. 📒⏩Comment Below If This Video Helped You 💯Like 👍 & Share With Your Classmates - ALL THE BEST 🔥Do Visit My Second Channel - https://bit.ly/3rMGcSAThis vi...1. Expand the function f(z) = 2(z + 2) z2 − 4z + 3 in a Taylor series about the point z = 2 and find the circle C inside of which the series converges. Find a Laurent series that converges in the region outside of C. I tried writing the denominator as (z − 3)(z − 1) to find the singularities z = 1, 3 both simple.Revising derivation in Taylor's series expansion.Go to http://www.examsolutions.net/ for the index, playlists and more maths videos on Taylors series...Yellowstone, the hit television series created by Taylor Sheridan and John Linson, has captivated audiences around the world. Known for its breathtaking landscapes and gripping sto...Taylor series are expansions of f (z) about points z 0, known as regular points, within the region where the function is analytic.It is also possible to expand a function about singular points. Fig. 15.3 outlines an annular (lock washer shaped) region around a singularity z 0 of a function f q (z), but avoiding other singularities at z 1 and z 2.The function is integrated …Applying Taylor expansion in Eq. (16), S 1s,1s is expanded at expansion center a 0 and b 0 as shown in the Appendix.The degree of approximation of S 1s,1s expressed in Taylor-series can be controlled by sliding expansion center, {a 0, b 0}, appropriately.It is possible to define an approximate Hamiltonian using such molecular integrals of controlled precision.Appendix A: Taylor Series Expansion. 221. In particular, it means that we only need to keep first-order terms and only one second-order term (dB dB × = dt), ignoring all other terms. Starting with. dX(t , ω) = μ(t , ω)dt. + σ(t , ω)dB(t, ω) we proceed formally with Taylor Series for a function of two variables f (t X), and ignore all ...Get ratings and reviews for the top 11 pest companies in Taylor, MI. Helping you find the best pest companies for the job. Expert Advice On Improving Your Home All Projects Feature...Taylor Series Expansion Calculator computes a Taylor series for a function at a point up to a given power. Taylor series is a way to represent a function as a sum of terms calculated based on the function's derivative values at a given point as shown on the image below. Taylor series calculator present the computed Taylor series as sum of its ...طريقة إيجاد صورة الرقم من خلال استخدام Taylor Series.Khaled Al Najjar , Pen&Paperلاستفساراتكم واقتراحاتكم :Email: khaled.civil95@gmail ....

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