Taylor's theorem gives an approximation of a n times differentiable function around a given point by a n-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are fixed order truncations of its Taylor's series, which completely determines the function in some locality of the point. There are numerous forms of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor-polynomial.
Some examples of Taylor’s theorem are:
Ex. 2) Expand log tanπ4+x in ascending orders of x.
Example 3: In order to write or calculate a Taylor series for we first need to calculate its n -derivatives, which we have already done above. The Taylor Series is defined as:
Simplifying it we get:
The easiest number to choose for a is probably 1, though you can choose whatever number you want to for a , so long as its n derivatives are all defined at a.
Substituting a for 1 gives:
Now let us evaluate f(x) at x=6 using the Taylor Series:
Relationship to Analyticity.
Taylor’s expansion of real analytical functions
Let I⊂R be an open interval. By definition, a function f:I→R is real analytic if it is locally defined by a convergent power series. This means that for every a ∈ Ithere exists some r > 0 and a sequence of coefficients ck ∈ R such that (a − r, a + r) ⊂ I and
In general, the radius of convergence of a power series can be computed from the Cauchy–Hadamard formula
This result is based on comparison with a geometric series, and the same method shows that if the power series based on a converges for some b∈R, it must converge uniformly on the closed interval [a − rb, a + rb], where rb = |b − a|. Here only the convergence of the power series is considered, and it might well be that(a − R,a + R) extends beyond the domain I of f.
The Taylor polynomials of the real analytic function f at a are simply the finite truncations