Finding Mclaurin polynomials

 Example

Solution

The graph of the function f and the first three polynomials are shown here:

We also have f⁵(x) = cosx and f⁵(0) =1

Let's find the first 3 Taylor polynomials and the 2 next subsequent ones in order to find a pattern allowing to find a formula for the nth Taylor polynomial.

We see here a pattern where p₁ = p₂,  p₃ = p₄ and p₅ = p₆. The Taylor polynomial of odd subscript is equal to the subsequent Taylor polynomial with even subscript.. If we call 2m+1 the odd subscript and 2m+2 the even subscript, we have p₂ₘ₊₁(x) = p₂ₘ₊₂(x) and:

The graphs of the function and its Mclaurin polynomials are shown here:

f⁵(0) = 0

Let's calculate the first 3 polynomials and the 2 subsequent ones in order to find a pattern that allows to write a formula for the nth Taylor polynomial.

Graph of the function and the Mclaurin polynomials: