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IT - 1ID NO:1 To 5Sub: Calculus

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About Brook Taylor

Brook Taylor was born in Edmonton on 18 August 1685

He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees of LL.B. and LL.D. in 1709 and 1714, respectively.

Having studied mathematics under John Machin and John Keill, in 1708 he obtained a remarkable solution of the problem of the "centre of oscillation,"

which, however, remained unpublished until May 1714, when his claim to priority was disputed by Johann Bernoulli.

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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the

function's derivatives at a single point.

The concept of a Taylor series was formulated by the Scottish mathematician James Gregory and formally introduced by the English

mathematician Brook Taylor in 1715.

If the Taylor series is centered at zero, then that series is also called aMaclaurin series

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A function can be approximated by using a finite number of terms of its Taylor series.

Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation. The polynomial formed by taking some initial terms of the Taylor series is called a

Taylor polynomial.

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 known as an analytic function in that interval.

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The Taylor series of a real or complex-valued function f(x) that is infinitely differentiable at a real or complex number 𝛼 is the power series

which can be written in the more compact sigma notation as

Where 𝑛! Denotes the factorial ofn and denotes the nth derivative of f evaluated at the point a.

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The nth Maclaurin polynomial for a function as :

nn

kn

k

k

xn

fx

fxffx

k

f

!

)0(...

!2

)0()0()0(

!

)0( 2//

/

0

The nth Taylor polynomial for f about x = x0 as :

nn

kn

k

k

xxn

xfxx

xfxxxfxfxx

k

xf)(

!

)(...)(

!2

)())(()()(

!

)(0

02

00

//

00

/

00

0

0

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If f(x) is given by a convergent power series in an open disc (or interval in the real line) centered at b in the complex plane, it is said to be analytic in this disc. Thus for x in this disc, f is given by a convergent power series

Analytic functions

Differentiating by x the above formula n times, then setting x=b gives

and so the power series expansion agrees with the Taylor series.

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EXAMPLE

Find the Maclaurin series of the function f(x) =ⅇ𝑥 and its radius of convergence.

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THEOREM (TAYLOR’S FORMULA)

EXAMPLE 4

EXAMPLE 4