Chapter 9.3, Problem 15E

To determine

To prove: The Maclaurin series for the given function converges to that the function for all real $x$ .

Explanation of Solution

Given information:

The function is $\cos x$ .

Calculation:

First let examine the derivative of $f(x)=\cos x$ .

Hence,

  $\begin{array}{c}{f}^{\prime }(x)=-\sin x\\ f^{″}(x)=-\cos x\\ f^{‴}(x)=\sin x\\ f^{(4)}(x)=\cos x\\ =f(x)\end{array}$

The further derivatives are identical to the aforementioned first four, so

  $\left|f^{(n+1)}(t)\right|\le 1$

For each

  $t\in ℝ$ , since let $M=1$

And

  $r=1\text{,}$

i.e.

  $\left|f^{(n+1)}(t)\right|\le M{r}^{n+1}$

for all $t$ between $a=0$ and $x$ , thus according to Theorem 4 the Maclaurin series for $\cos x$ converges for all real $x$ .

The convergence can be proved by using the Remainder Bounding Theorem with $M=1$ and

  $r=1$ .