The Taylor series provides a means to approximate a function value at one point based on the function value and its derivatives at another known point. It states that any smooth function can be approximated as a polynomial. The Taylor series expansion allows estimating the value of a function like x^100 at a point like x=20 by using the known value and derivatives of the function at another point, like x=1. Increasing the order of the Taylor series approximation or decreasing the step size between points improves the accuracy of the approximation.

1. Taylor series
The tailor series provides a means to predict a function value at one point in
terms of the function value and its derivatives at another point.
The theorem states that any smooth function can be approximated as a
polynomial.
To understand: if u want to determine value of x^100 at x=20. You know
suppose, at x=1 x^100=1. Now with this point (x=1) by using Taylor series you can
determine the value of x^100 at x=20 or others.
How?
Suppose at x=a you know the value of the given function.
From Taylor expansion:
( )∗ ( )∗ ( )∗
( ) = ( )+ ( )∗ℎ+ + + ......+ +
! ! !
Here, a= the known point. For this example a=1, f(a) =1.
h=x-a. Where at x=x f(x) wanted to determine. In this example x=20.
Rn = Remainder.
Its need infinite term for 100% accuracy.
But as its not possible we cut the series in a significant figure say n. it is called nth
order equation.
To compensate we add a remainder Rn for the remaining term.
If n is equal to the actual order of the analytical equation than Rn is not needed.

2. Effects of step size:
Suppose, ( )= at x=1 f(1)=1. Now we want to know f(2) =?
True value: f(2)=2^4=16
If expand with Taylor series,
ℎ
( + 1) = + 4 ∗ ∗ℎ+6∗ ∗ ℎ + 2 ∗ ∗ℎ +
12
Now, h=1 and X i+1=2, so x= X i+1 – h= 2-1 =1
f(2)=1+4+6+2+1/12=13.08333
et= ( (16 – 13.08333) / 16) * 100% = 18.229%
Now h=0.5 and X i+1=2, so x= X i+1 – h = 2 – 0.5 =1.5
f(2) = 15.5677
et=2.7018%
Now h=0.25 and X i+1=2, so x= 1.75
F(2)=15.9417
et=0.036%
look, as we decreasing step size true error is also decreasing. That means if step
size is smaller, accuracy will be higher.

3. Effect of order:
Again consider ( ) =
At X i+1 =2 and h = 0.5 we want to determine f(x) by adding term one after one.
So xi=1.5
If we expand with Taylor series zero order approximation,
( + 1) =
f(2)=5.0625
et = 68.3594%
if we expand with first order approximation,
( + 1) = + 4 ∗ ∗ℎ
f(2) = 11.8125
et = 26.1719%
again expand with 2nd order approximation,
( + 1) = + 4 ∗ ∗ℎ+6∗ ∗ℎ
f(2) = 15.1875
et = 5.078%
This says that, if we add more terms or increase order, result will goes to close to
the true value.