I am self studying calculus and I need help solving a Taylor Series problem.
1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0:
I think the answer would be:
1-(x^2/2!)+(x^4/4!)-(x^6/6!)+(x^8/8!)
Not entirely sure though.
1b) Approximate cos(x) with P{4}(x) to simplify the ratio: (1/cos(x)) /x
I am unsure how to approach and solve this part of the problem.
1c)Using this, conclude the limit as x goes to infinity
(1/cos(x)) /x
I believe the answer to this would be 0. It is an identity in the book.
Overall, I am not sure if I am going in the right direction with this. Some help and guidance would be much appreciated!
Taylor's Formula is given by $$\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$$ where $a$ is where $f^{(n)}$ is evaluated at.
For 1a), you need to calculate this sum up to $n=4$, not $n=8$. That is $\sum_{n=0}^4\dfrac{f^{(n)}(a)}{n!}(x-a)^n.$ This will be $P_4[x].$
For 1b), you know from 1a) that you can approximate $\cos(x)$ with $P_4[x].$ Then, you will have $\dfrac{1}{x\cos(x)}\approx\dfrac{1}{xP_4[x]}.$
For 1c), then all you need to do is take $\lim\limits_{x\to\infty}\dfrac{1}{xP_4[x]}.$ Hint You can simplify it to end up with something of the form $\dfrac{c}{P[x]},$ where $c$ is a constant. Since $P[x]$ grows without bound, then you will end up with a constant divided by an increasingly large number. Therefore, the result will become increasingly small. i.e, we will get $0$ in the limit.