I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will be posting a lot of questions with the exercises I find challenging, and I would like to ask for any help or methodologies that will make it easier for me to solve.
I understand the whole ordeal is categorized as "homework", but any assistance would be appreciated, as I'm completely clueless and I would like to be prepared.
The following exercise is $Ex. 10$, graded for $10\%$.
Define Taylor's Theorem and prove that $\sin(x) = \sum\limits_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$.
From the class'es page, I've got the definition of Taylor's theorem along with the following equation that resembles a Sum:
$$f(x)=f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+\frac{f^{(n+1)}(x_0)}{(n+1)!}(x-x_0)^{n+1}$$
Apart from the similarity, I do not know how to continue. Do I take the second part of the equation and try to find a closed type that leads to sin(x)?
Answer
Notice that
$$f(x) = sin x \ \ \ \ \ \ \ \ f(0)= 0\\ f'(x) = cos x \ \ \ \ \ \ f'(0)=1 \\ f''(x) = -sinx \ \ f''(0)=0 \\ f'''(x) = -cosx \ \ f'''(0) = -1 \\ f^{(4)}(x)=sinx \ \ \ f(0)=0 $$
The derivatives repeat in a cycle of four, from Taylor series definition we have that $$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)(x-a)^{n}}{n!}$$ Take $a=0$ and the result follows.
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