Math Help - Taylor's Theorem problem 2

1. Taylor's Theorem problem 2

This involves a proof to Taylor's Theorem. Assume $x >0$ and let $f$ be defined on $(a,b)$ where $a < 0 < b$ and suppose the n-th derivative $f^{(n)}$ exists on $(a,b)$. Then for each nonzero $x\in (a,b)$ there is some $y$ between $0$ and $x$ such that $R_{n}(x)=\frac{f^{(n)}(y)}{n!}x^{n}$.

Let $M$ be the unique solution of $f(x)= \sum_{k=0}^{n-1}\frac{f^{(k)}(0)}{k!}x^{k} + \frac{Mx^{n}}{n!}$ and let $F(t) = f(t) + \sum_{k=0}^{n-1}\frac{(x-t)^{k}}{k!}f^{(k)}(t) + M\frac{(x-t)^{n}}{n!}$

Show that $F$ is differentiable on $[0,x]$ (I think this part is straightforward since $f$ is differentiable on $(a,b)$ and the other terms are simply polynomials). Show that $F'(t)= \frac{(x-t)^{n-1}}{(n-1)!}[f^{(n)}(t) - M]$ (not really sure how this works out, not great with combining series), show that $F(0)=F(x)$ (again, don't know), and finally show that there is $y\in (0,x)$ such that $f^{(n)}(y) = M$.

My professor showed an alternate proof in class which seems at least somewhat easier, but I am absolutely lost here. Thanks.

2. When you differentiate, the sum in F telescopes (use the product rule). To show $F(0) = F(x)$, just, uh, plug those in.

3. Okay, I forgot the first time of the series doesn't cancel out when you plug in x, so that was throwing me off. Like I said, infinite series was my weakest area in basic calculus and now in analysis.

Ugh, nevermind, I'm not getting equal answers after all. I'm getting $F(0) = f(x) + (0)$ and $F(x) = 2f(x)$. I don't know what I'm doing wrong....

4. How did you get F(x)=2f(x)?

5. Er, I plugged in a $k=0$, not $k=1$ first for the series. Okay, so I'm hoping this is correct:

$F(0) = f(0) + f(x) - f(0)$; the last two terms are pretty much $f(x)$ except the series is from 1 to n-1, not 0 to n-1, so we have to subtract a $f(0)$ to compensate.

$F(x) = f(x) + 0$ and so both are equal to $f(x)$, yes?

6. Since M is chosen so that $f(x)= \sum_{k=0}^{n-1}\frac{f^{(k)}(0)}{k!}x^{k} + \frac{Mx^{n}}{n!}
$
, just see that F(0) is the RHS of this.

So, ja, they're both f(x).

7. Okay, thanks after that I can take it from there with the Rolle's and all that. For some reason when it comes to series my mind hits a brick wall.