Approximating sinx using Mclaurin polynomials

 

The Mclaurin polynomials allow to approximate a function. Let's use them to find approximations of sinx

Example

From the example in the blogpost "Finding Mclaurin polynomials", the Mclaurin polynomials for sinx are given by:

Solution

Let's estimate the error. The sixth Mclaurin polynomial is equal to the fifth Mclaurin polynomial : p₆(x) = p₅(x). Let's calculate a bound on R₆ (Π/8). Since the Taylor series is unique, the remainder is:

b) The remainder on the sixth Taylor polynomial is given by the formula:

┃Rₙ(x)│≤ M/(n+1)!❘│x│⁷

Substituting the letters by their value:

0.0001 ≤ 1/7! │x│⁷

Solving this inequality we find

│x│≤ 0.907