Originally Posted by

**earboth** Hello,

I would draw a rough sketch of the 2 graphs. For instance with #1:

Draw the graph $\displaystyle f(x)=2\cos(x)\ ,~-\pi < x < \pi$ anf $\displaystyle p(x)=x^4$. Use the estimated x-value of the intersection as an initial value.

When I used $\displaystyle x_0 = 1$ I got the result with 8 decimals exact after 3 steps.

to #2. If n is a large number then $\displaystyle \sqrt[n]{n} \approx 1$. If you use a value a little bit larger than 1 as an initial value it will be sufficient. But with your example the sequence of x-value converges very slowly. So when I used $\displaystyle x_0 = 1.5$ it took me more than 20 cycles to get a nearly exact value.