Taylor's theorem states that any function satisfying certain conditions can be expressed as a Taylor series. A Taylor series is a series expansion of a function about a point, giving an approximation of the function near that point. The Taylor series for a function y(x) around a point x=x0 is given by y1 = y0 + (x-x0)/1! * y0' + (x-x0)2/2! * y0'' + (x-x0)3/3! * y0''' + ..., providing successive approximations of the function near x0 using derivatives of the function evaluated at x0. Similarly, the Taylor series can be developed around any point x=x1

1. Government Engineering
College - Modasa
Name : Milan Bhatiya
Subject : Numerical and Statistical Methods
Topic : Taylor’s Series Method

2. TAYLOR SERIES THEORY
It is common practice to use a finite number of terms of the
series to approximate a function. The Taylor series may be regarded as
the limit of the Taylor polynomials.
Taylor's theorem (actually discovered first by Gregory) states
that any function satisfying certain conditions can be expressed as
a Taylor series.
A Taylor series is a series expansion of a function about a
point. A one-dimensional Taylor series is an expansion of a real
function about a point is given by

3.  𝑦1 = 𝑦0 +
𝑥−𝑥0
1!
𝑦0
′
+
𝑥−𝑥0
2
2!
𝑦0
′′
+
𝑥−𝑥0
3
3!
𝑦0
′′′
+ ⋯ ….(1)
Putting 𝑥 − 𝑥0 = h in EQ.1
 𝑦1 = 𝑦0 +
ℎ
1!
𝑦0
′
+
ℎ2
2!
𝑦0
′′
+
ℎ3
3!
𝑦0
′′′
+ ⋯ …..(2)
TAYLOR SERIES THEORY

4. Similarly, Taylor series for y(x) around x=x1 is given by
 𝑦2 = 𝑦1 +
ℎ
1!
𝑦1
′
+
ℎ2
2!
𝑦1
′′
+
ℎ3
3!
𝑦1
′′′
+ ⋯
proceeding in the same way,
 𝑦 𝑛+1 = 𝑦𝑛 +
ℎ
1!
𝑦𝑛
′ +
ℎ2
2!
𝑦𝑛
′′ +
ℎ3
3!
𝑦𝑛
′′′ + ⋯
TAYLOR SERIES THEORY