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Newton Raphson iteration

CasualSoul

(please see the two pages attatched) - having bit of trouble understanding one thing

The first bit is just a mistake, yes.

For the second bit, if you understand how the method works, just try to make a counterexample, where the iterations don't get closer to the desired root. An obvious problem is that there could be multiple roots and you overshoot it and then start closing on the wrong one. If you have a minimum before the root, you could also end up in a 'well'. I haven't studied the general convergence of these things, but it could be quite complicated. The only thing it wants you to realise is that you have to get 'close enough' to the root so there aren't any problems inbetween.