Quantcast
Channel: Cannot understand calculations (Taylor's Theorem) - Mathematics Stack Exchange
Viewing all articles
Browse latest Browse all 4

Cannot understand calculations (Taylor's Theorem)

0
0

I have been studying Taylor's Theorem by Bartle's book. However, i cannot understand some simple(?) calculations, and that is really bothering me.

Theorem (as stated in the book):

Let $n \in \mathbb{N},$ let $I:=[a, b],$ and let $f: I \rightarrow \mathbb{R}$ be such that $f$ and its derivatives $f^{\prime}, f^{\prime \prime}, \ldots, f^{(n)}$ are continuous on I and that $f^{(n+1)}$ exists on $(a, b)$. If $x_{0} \in I,$ then for any $x$ in I there exists a point $c$ between $x$ and $x_{0}$ such that:$$f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+\frac{f^{\prime \prime}\left(x_{0}\right)}{2 !}\left(x-x_{0}\right)^{2}$$$$+\cdots+\frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}+\frac{f^{(n+1)}(c)}{(n+1) !}\left(x-x_{0}\right)^{n+1}$$

Proof:

Let $x_{0}$ and $x$ be given and let $J$ denote the closed interval with endpoints $x_{0}$ and $x$. We define the function $F$ on $J$ by:$$F(t):=f(x)-f(t)-(x-t) f^{\prime}(t)-\cdots-\frac{(x-t)^{n}}{n !} f^{(n)}(t)$$for $t \in J$. Then an easy calculation shows that we have:$$F^{\prime}(t)=-\frac{(x-t)^{n}}{n !} f^{(n+1)}(t)$$

The proof continues here, but the rest is fine, i got it.

What i cannot understand is the following:

Then an easy calculation shows that we have:$$F^{\prime}(t)=-\frac{(x-t)^{n}}{n !} f^{(n+1)}(t)$$

There are a lot of terms that just seem to have vanished from the expression regarding $F(t)$ and i just dont know how that happened. Furthermore, the last term $$F^{\prime}(t)=-\frac{(x-t)^{n}}{n !} f^{(n+1)}(t)$$

does not make sense to me.

What i got from the derivative of the last term $-\frac{(x-t)^{n}}{n !} f^{(n)}(t)$ was:$$\left((x-t)^{n} \cdot f^{(n)}(t)\right)^{\prime}=n(x-t)^{n-1} \cdot(-1) \cdot f^{(n)}(t)-(x-t)^{n} \cdot f^{(n+1)}(t)$$

Now, dividing by $n!$ what i got was:

$$\frac{-n(x-t)^{n-1} \cdot f^{(n)}(t)}{n \cdot(n-1) !}-\frac{(x-t)^{n} \cdot f^{(n+1)}(t)}{n !}$$

which does not resemble not even the last term regarding the $F(t)$ expression.

Can someone help?Thanks in advance, Lucas


Viewing all articles
Browse latest Browse all 4

Latest Images

Trending Articles





Latest Images