Taylor Series

1. Taylor Series • Brook Taylor • Presented By: Sumranayasmeen • Topic: Taylor series • Branch: HNS • Subject: Mathematics 1V • Year: 2 • Semester: 1

2. History of Taylor Series • Brook Taylor • Born:18 August 1685 in Edmonton, Middlesex, England • Died:29 December 1731 in Somerset House ,London , England. • The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. • Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. • It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later.

3. In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century. • In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. after whom the series are now named. • The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century

4. Introduction • In Mathematicians, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. if the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin who made extensive use of this special case of Taylor’s series in the 18th century. It is common practice to use a finite number of terms of the series to approximate a function. Taylor series may be regarded as the limit of the Taylor polynomials. • Taylor’s theorem gives quantitive estimates on the error in this approximation. Any finite number of initial terms of the Taylor polynomial. The Taylor series of a function is the limit of that function’s Taylor polynomials, provide that the limit exists. A function may not be equal to its Taylor series, even point. A function that is equal to its Taylor series in an open interval or a disc in the complex plane) is known as an analytic function.

5. Definition  If (f ) is defined in the interval containing “a” and its derivatives of all orders exist at x=a, then we can expand f(x) as f(x)= Which can be written in the more compact sigma notation as Where n! denotes the factorial of n and f(n)(a) denotes the nth derivative of f evaluates at the point a. the derivative of order zero f is defined to be 1.in the case that a=0.

6. Taylor series is not valid if anyone of the following holds • 1) At least one of f, f’, f’’,….f(n) becomes infinite on +a, a+h* • 2) at least on of f, f’, f’’,……f(n) is discontinuous on +a, a+h* • 3) limn ∞ Rn=0

7. Uses Of Taylor Series For Analytic Functions Include • The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included. • Differentiation and integration of power series can be performed term by term and is hence particularly easy. • An analytic function is uniquely extended to a holomorphic function on an open interval in the complex plane. This makes the machinery of complex analysis available. • Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach if often used in physics.

8. Example 1 Q. Find the Taylor series expansion of ln(1+x) at x=2 Solution: Let f(x)=ln(1+x) then f(2)=ln(1+2)=ln3 Finding the successive derivatives of ln(1+x) and evaluating them at x=2 f ꞌ(x)=1/1+x f ꞌ(2)=1/1+2=1/3 f ꞌꞌ (x)=(-1)(1+x)^-2 f ꞌꞌ(2)=-(1+2)^-2=-1/9 f ꞌꞌꞌ(x)=(-1)(-2)(1+x)^-3 f ꞌꞌꞌ(2)= 2 .(1+2)^-3= 2/27 The Taylor series expansions of f at x꞊a is

9. f(x)= Now substituting the relative value Ln(1+x) =ln3+1/3(x-2)+(-1/9)/2 (x-2) ² +(2/27)/3 (x-2) ³+……. =ln3+(x-2)/3-(x-2)²/9*2+2(x-2)³/162+…. =ln3+(x-2)/3-(x-2)²/18+(x-2)³/81+…….

10. Example 2 Sin 31⁰ A=30 ⁰ = π/6 Let F(x)=sinx F(π /6)=sin π /6 F(π /6)=1/2 Now taking the successive derivative of sinx and evaluating them at π /6. f ꞌ(x)=cosx f ꞌ (π /6)=cos(π /6)=√-3/2 f ꞌꞌ(x)=-sinx f ꞌꞌ (π /6)=-sinx(π /6)= -½

11. F ꞌꞌꞌ (x)=-cosx f ꞌꞌꞌ(π /6)=-cosx(π /6)=-√ 3/2 f⁴(x)=-(-sinx)=sinx f⁴(π /6)=sinx π /6=½ Thus the Taylor series expansion at a= π /6 f ꞌ (x)= Sinx=½+ √ 3/2 (x- π /6)+(-½)(x- π /6)/2+(-√ 3/2 )(x- π /6)/3+… for x=31⁰ x- π /6=(31⁰ -30⁰)=1⁰ =.017455 Sin 31⁰=½+√ 3/2(.017455)-1/4(.017455)-√ 3/2(.017455)+… ≈.5+.015116-0.00076 sin31⁰ ≈ .5156

12. Example 3 Q. Using Taylor’s Theorem to prove that lnsin(x+h)=lnsinx+hcotx1/2h²csc²x+1/3h³cotxcsc²x+… Solution: Let f(x+h)=lnsin(x+h) let x+h=x f(x) = lnsinx f (x) =1/sinx.cosx=cotx f (x) = -csc²x f (x) = -2cscx(-cscx.cotx) = 2csc²xcotx By Taylor’s Theorem, we get f(x+h)= f(x)+f (x)h/1 +f (x)h²/2 +…………….. lnsin(x+h) = lnsinx+hcotx+h²/2 (-csc²x)+h³/3 (2csc²xcotx)+… =lnsinx+hcotx-h²/2csc²x+h³/3csc²xcotx+…. Hence it is proved……..

13. Application Of Taylor Series In this section we will show you a few ways in Taylors Series which helps you to solve problems easily. • To find sum of series. • To evaluate limits. • It is used to approximate polynomials function.

14. Thank You