Brook Taylor was an English mathematician who formally introduced the Taylor series in 1715. A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It can be used to approximate functions by taking a finite number of terms of the Taylor series. The Taylor theorem gives quantitative estimates on the error of such approximations. A function is analytic if it is equal to its Taylor series in an open interval, meaning it can be expressed entirely as the Taylor series within that interval.

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

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About Brook Taylor
 BrookTaylor 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which, however, remained unpublished until May 1714, when his claim to priority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, aTaylor 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 aTaylor series was formulated by the Scottish
mathematician James Gregory and formally introduced by the English
mathematician BrookTaylor in 1715.
If theTaylor series is centered at zero, then that series is also called a
Maclaurin series

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 A function can be approximated by using a finite number of terms of
itsTaylor 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 theTaylor series is called a
Taylor polynomial.
 TheTaylor series of a function is the limit of that function'sTaylor polynomials as the
degree increases, provided that the limit exists.
 A function may not be equal to itsTaylor series, even if itsTaylor series converges at every
point. A function that is equal to itsTaylor 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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TheTaylor 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 of
n and denotes the nth derivative of f evaluated at the point a.

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 The nth Maclaurin polynomial for a function as :
n
n
k
n
k
k
x
n
f
x
f
xffx
k
f
!
)0(
...
!2
)0(
)0()0(
!
)0( 2
//
/
0

 The nth Taylor polynomial for f about x = x0 as :
n
n
k
n
k
k
xx
n
xf
xx
xf
xxxfxfxx
k
xf
)(
!
)(
...)(
!2
)(
))(()()(
!
)(
0
02
0
0
//
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) =ⅇ 𝑥 and its radius of convergence.

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THEOREM (TAYLOR’S FORMULA)

13. EXAMPLE 4

14. EXAMPLE 4