We are familiar with polynomial equations of the form f(x) = a_{0} + a_{1}x + . . . + a_{n} x^{n} where a_{0}, a_{1}, . . . , a_{n} are real numbers. We can easily compute the value of a polynomial at any point x = a by using the four basic operations of addition, multiplication, subtraction and division. On the other hand there are functions like e^{x}, cos x, ln x etc. which occur frequently in all branches of mathematics which cannot be evaluated in the same manner. For example, evaluating the function f(x) = cos x at 0.524 is not so simple. Now, to evaluate such functions we try to approximate them by polynomials which are easier to evaluate. Taylor's theorem gives a simple method for approximating functions f(x) by polynomials.

Let f(x) be a real-valued function defined on R which is n-times differentiable.

Consider the function where x

_{0} is any given real number. Now P

_{l}(x) is a polynomial in x of degree 1 and P

_{1} (x

_{0}) = f(x

_{0}) and P

_{1} '(x

_{0}) = f '(x

_{0}). The polynomial P

_{1}(x) is called the first Taylor polynomial of f(x) at x

_{0}.
Now consider another function

Then P

_{2}(x) is a polynomial in x of degree 2 and P

_{2}(x

_{0}) = f(x

_{0}), P '

_{2}(x

_{0}) = f'(x

_{0}) and P ''

_{2}(x

_{0}) = f '(x

_{0}). P

_{2}(x) is called the second Taylor polynomial of f(x) at x

_{0}.

Similarly we can define the rth Taylor polynomial of f(x) at x

_{0} where 1 ≤ r ≤ n. The rth Taylor polynomial at x

_{0} is given by

## Theorem

Let f be a real valued function having (n + 1) continuous derivatives on ]a, b[ for some n ≥ 0. Let x

_{0} be any point in the interval ]a, b[. Then for any x ∈ ]a, b[, we have

where c is a point between x

_{0} and x.

The series is called the nth Taylor's expansion of f(x) at x

_{0}.

## Remainder or error

We can write the above equation as

where P

_{n}(x) is the nth Taylor polynomial of f(x) about x

_{0} and

R

_{n+1}^{(x)} depends on x, x

_{0} and n. R

_{n+1}(x) is called the remainder (or error) of the nth
Taylor's expansion after n + 1 terms.

Suppose we put x

_{0} = a and x = a + h where h > 0 . Then any point between a and a+h will be of the form a+θh,O<θ<1.

Therefore we can write the equation as