Question

Question: To determine the Taylor polynomials converge to\(f(x)\).

Short answer

The Taylor polynomials \({T_0}(x),{T_2}(x),{T_4}(x),{T_6}(x)\) and the given function \(f(x)\) are converges at \(x = 0\).

Step-by-step solution

Given data

The Taylors polynomials are \({T_0}(x) = 1,{T_2}(x) = 1 - \frac{{{x^2}}}{2},{T_4}(x) = 1 - \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}}\) and \({T_6}(x) = 1 - \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}}\). \( - \frac{{{x^6}}}{{720}}\).

Concept used of the \(n\) th-degree of the Taylor polynomial

The\(n\)th-degree of the Taylor polynomial of\(f(x)\)at\(a\)is\({T_n}(x) = \sum\limits_{i = 0}^\infty {\frac{{{f^{(i)}}(a)}}{{i!}}} {(x - a)^i}\),

\({T_n}(x) = f(a) + \frac{{{f^\prime }(a)}}{{1!}}(x - a) + \frac{{{f^{\prime \prime }}(a)}}{{2!}}{(x - a)^2} + \frac{{{f^{\prime \prime \prime }}(a)}}{{3!}}{(x - a)^3} + \cdots {\rm{ }}\)

Draw the graph

The Graph is,

Analyse the graph

From the Figure part (a) it is clearly observed that, the Taylor polynomials \({T_0}(x),{T_2}(x),{T_4}(x),{T_6}(x)\) gives the closer result to the given function \(f(x)\) when \(x\) tending to zero.

Therefore, the Taylor polynomials \({T_0}(x),{T_2}(x),{T_4}(x),{T_6}(x)\) and the given function \(f(x)\) are converges at \(x = 0\).