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### Taylor's Theorem

We are familiar with polynomial equations of the form f(x) = a0 + a1x + . . . + an xn where a0, a1, . . . , an 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 ex, 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 x0 is any given real number. Now Pl(x) is a polynomial in x of degree 1 and P1 (x0) = f(x0) and P1 '(x0) = f '(x0). The polynomial P1(x) is called the first Taylor polynomial of f(x) at x0. Now consider another function

Then P2(x) is a polynomial in x of degree 2 and P2(x0) = f(x0), P '2(x0) = f'(x0) and P ''2(x0) = f '(x0). P2(x) is called the second Taylor polynomial of f(x) at x0.

Similarly we can define the rth Taylor polynomial of f(x) at x0 where 1 ≤ r ≤ n. The rth Taylor polynomial at x0 is given by

## Theorem

Let f be a real valued function having (n + 1) continuous derivatives on ]a, b[ for some n ≥ 0. Let x0 be any point in the interval ]a, b[. Then for any x ∈ ]a, b[, we have

where c is a point between x0 and x.

The series is called the nth Taylor's expansion of f(x) at x0.

## Remainder or error

We can write the above equation as

where Pn(x) is the nth Taylor polynomial of f(x) about x0 and

Rn+1(x) depends on x, x0 and n. Rn+1(x) is called the remainder (or error) of the nth Taylor's expansion after n + 1 terms.

Suppose we put x0 = 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