Can you solve it by tapping the same function key?

Calculus Level 4

Do you remember a time when you got that first calculator with the extra functions?

I remember entering a value and hitting the cosine key for example until the numbers stopped changing.

That is I solved $\cos{x^*} = x^*$ by iteration:

Suppose I started with $x_1 = 1$ , and I hit the cosine key (in radians mode). I get 0.5403. Plug this back into cosine and do it all over again.

The idea is $x_{n+1} = \cos(x_n)$ .

Suppose as $N\rightarrow \infty$ we find $|\cos(x_m)-\cos(x_n)|\leq k|x_m-x_n|^q$ for some $n$ , $m > N$

Find the value of $k+q$ .

The answer is 1.673612.

1 solution

Abhishek Sinha
May 22, 2019

As stated in the problem (which can be formally proved by Banach's fixed point theorem) $x_n \to x^*$ , where $x^*$ is the solution of the fixed point equation $x^*=\cos(x^*).$ Numerically, $x^* \approx 0.739085.$

Now, for large enough $N$ , we write $|\cos(x_m)-\cos(x_n)|=2|\sin(\frac{x_m+x_n}{2})||\sin(\frac{x_n-x_m}{2})| \stackrel{(a)}{\leq} \sin(x^*) |x_m-x_n|,$ where the inequality (a) follows from the fact $\sin(z) \leq z$ and $x_m, x_n \to x^*$ .

Thus, we conclude that $k=\sin(x^*) \approx 0.673612$ and $q=1$ .

Can you solve it by tapping the same function key?

The answer is 1.673612.

1 solution

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