Hi, could anyone help me with part b of this question, part a I have completed, however I seem to be drawing a blank on the second part
Hint: Just use the definition of the derivative to get
$\displaystyle \displaystyle f'''(x)=\lim_{h \to 0}\frac{f''(x+h)-f''(x)}{h}= -\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=-f,(x)$
Now use the above to find the forth derivative then you can probe that
$\displaystyle f^{n+4}(x)=f^{n}$ for all $\displaystyle n \in \mathbb{N}$
Then use this relationship to find the Taylor series for this function.
Just FYI this is the function $\displaystyle \cos(x)$