The document discusses the history and development of Taylor series. Some key points: 1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point. 2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory. 3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.

2. TAYLOR
SERIES

3. 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
exhaustionthat an infinite number of progressive subdivisions
could be performed to achieve a finite result.[1] Liu
Hui independently employed a similar method a few centuries
later.[2]

4. • In the 14th century, the earliest examples of the use of Taylor
series and closely related methods were given by Madhava of
Sangamagrama.[3][4] 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,[5] 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.

5. 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 Maclurin
series, named after the Scottish mathematician Colin
Macluarin 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.

6. 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.

7. 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.

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

9. Uses of Taylor series for
analytic functions include
1. 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.
2. Differentiation and integration of power series can
be performed term by term and is hence
particularly easy.
3. 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.

10. 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.

11. Example 1
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 (x)=1/1+x
f (2)=1/1+2=1/3
f (x)=(-1)(1+x)^-2
f (2)=-(1+2)^-2=-1/9
f (x)=(-1)(-2)(1+x)^-3
f (2)= 2 .(1+2)^-3= 2/27
The taylor series expansions of f at x=a is

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

13. Example no.2
Sin 31⁰
A=30 ⁰=/6
Let
F(x)=sinx
F(/6)=sin /6
F(/6)=1/2
Now taking the successive derivative of sinx and
evaluating them at /6.we
f (x)=cosx
f (/6)=cos(/6)=√
f (x)=-sinx
f (/6)=-sinx(/6)=-½

14. F’’’(x)=-cosx
f’’’(/6)=-cosx(/6)=-√
f⁴(x)=-(-sinx)=sinx
f⁴(/6)=sinx /6=½
Thus the Taylor series expansion at a= /6
f (x)=
Sinx=½+ √ (x- /6)+(-½)(x- /6)/2+(-√ )(x- /6)/3+…
for x=31⁰
x- /6=(31⁰ -30⁰)=1⁰ =.017455
Sin 31⁰=½+√(.017455)-(.017455)-√(.017455)+…
≈.5+.015116-0.00076
sin31⁰ ≈ .5156

15. Example # 3
Using Taylor’s Theorem to prove that
lnsin(x+h)=lnsinx+hcotx-1/2h²csc²x+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……..

16. • 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.