# Thread: Taylor series analysis question

1. ## Taylor series analysis question

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

2. Originally Posted by LHS

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

$\displaystyle f'''(x)=\lim_{h \to 0}\frac{f''(x+h)-f''(x)}{h}= -\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=-f,(x)$

Now use the above to find the forth derivative then you can probe that

$f^{n+4}(x)=f^{n}$ for all $n \in \mathbb{N}$

Then use this relationship to find the Taylor series for this function.

Just FYI this is the function $\cos(x)$

3. Ah thank you, probably should have seen that, but that's what ten hours and about a gram of caffeine do to you. Cheers