example of Taylor polynomials for $sinx$

In this entry we compute several Taylor polynomials for the function $\sin x$ around $x=0$ and we produce graphs to compare the function with the corresponding Taylor polynomial. Recall that for a given function $y=f(x)$ (here we suppose $f$ is infinitely differentiable) and a point $x=a$, the Taylor polynomial of degree $n$ ($n\ge 0$) is given by:

$$T_n(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}{(x-a)}^2+\mathrm{\dots }+\frac{f^{(n)}(a)}{n!}{(x-a)}^n$$

where $f^{(n)}$ denotes the $n$th derivative of $f(x)$.

From now on we assume $f(x)=\sin x$ and $a=0$. Notice that the derivatives of $\sin x$ are cyclic:

$$f^{\prime }(x)=\cos x,f^{\prime \prime }(x)=-\sin x,f^{\prime \prime \prime }(x)=-\cos x,f^{(4)}(x)=\sin x=f(x).$$

Therefore, the Taylor polynomials are easy to compute. In fact:

$$f^{(2n)}(0)=0,f^{(2n+1)}(0)={(-1)}^n$$

Thus, the first Taylor polynomial is given by:

$$T_1(x)=0+1\cdot x=x$$

In the following graph one can compare the function $T_1(x)=x$ and $\sin x$.

The function $y=\sin x$ and the first Taylor polynomial.

Notice that $T_2(x)=T_1(x)$. More generally, $T_{2n}(x)=T_{2n-1}(x)$ so we will not compute any other even order Taylor polynomials. However, the third degree Taylor polynomial is given by the formula:

$$T_3(x)=x-\frac{x^3}{3!}=x-\frac{x^3}{6}$$

The function $y=\sin x$ and the third Taylor polynomial.

The Taylor polynomial of degree $5$ is given by:

$$T_5(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}$$

The function $y=\sin x$ and the fifth Taylor polynomial.

Next, we compute some Taylor polynomials of higher degree. In particular, the Taylor polynomial of degree $15$ has the form:

$$T_{15}(x)=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\frac{x^9}{362880}-\frac{x^{11}}{39916800}+\frac{x^{13}}{6227020800}-\frac{x^{15}}{1307674368000}$$

The function $y=\sin x$ and the Taylor polynomial of degree $15$.

Finally, we produce a detailed view of the Taylor polynomial of degree $99$. In particular, notice that the graphs are very close until $x=34$ or so, but after that $T_{99}(x)$ behaves rather jittery and wildly.

A detail of the Taylor polynomial of degree $99$ (the interval $(34,39)$).