# Taylor’s Theorem

**Taylor’s Theorem:**

Taylor's Theorem for a function of a single variable and Maclaurin's series functions. Suppose a function f (x) satiesfy the following conditions:

**1**. f (x), f' (x), f'' (x),........ f^{ (n-1)} (x) are continuous in the closed intervals [a,b]

**2**. f^{ (n-1)} (x) is differentiable i.e f^{n} (x) exists in the open intervals (a,b)

Then there exists a point c ∈ (a,b) such that;

Taylor's Theorem is more usually written in the following forms. Substitute b = x in the above equations we get;

Thus f (x) can be expressed as the sum of infinite series. This series is called the Taylor's series for the function f (x) about the point a. If we substitute a = 0 we get;

This is called Macluarin's series for the function f (x). If f (x) = y and f' (x), f'' (x), .................. are denoted by y_{1}, y_{2,} ................... The Macluarin's series can also be written in the form:

**Rolle’s Theorem:**

If a function f (x) is:

**1**. Continuous in a closed interval [a, b],

**2**. Differentiable in the open interval (a, b) and

**3**. f (a) = f (b). Then there exists at least one value c of x in (a, b) such that f ′ (c) = 0