The difference between Taylor seires in Evan's PDE book and Lee's intro to smooth manifolds I am not an expert in Taylor series, and I would like to learn more about it. But I am confused about two versions of multivariate Taylor series that I found. The first one is in Lawrence Evan's "Partial Differential Equations", which reads:
$f(x) = \sum\limits_{|\alpha| \leq k} \frac{1}{\alpha!}D^\alpha f(0)x^\alpha + O(|x|^{k+1}) \text{ as } x \to 0$
And the other version is what I found in John M. Lee's "Introduction to Smooth Manifolds", which reads:
$f(a) + \sum\limits_{m=1}^k\frac{1}{m!}\sum\limits_{I:|I|=m}\partial_I f(a)(x-a)^I+ R_k(x)$,
where $R_k(x) = \frac{1}{k!}\sum\limits_{I:|I|=k+1}(x-a)^I \int\limits^1_0 (1-t)^k \partial_If(a+t(x-a)) dt$
Here, as you see, there is no mention of the behavior of $R_k$ as $x\to a$. 
There is an obvious difference in notations, but it is not what causing the problem for me. The quantity $\frac{1}{\alpha!} \neq \frac{1}{k!}$ (at least not necessarily). 
Both versions seem to be true, but are they the same and I am missing something? 
I would highly appreciate it if you answer this for me.
 A: The difference is that Evans and I use different multi-index conventions. For Evans, if he's working in $\mathbb R^n$, a multi-index $\alpha$ always represents an $n$-tuple of nonnegative integers. If $\alpha = (\alpha_1,\dots,\alpha_n)$, then 
\begin{align*}
x^\alpha &= (x_1)^{\alpha_1}\dots (x_n)^{\alpha_n},\\
D^\alpha &= \frac{\partial^{\alpha_1 + \dots + \alpha_n}}{\partial x_1^{\alpha_1}\dots \partial x_n^{\alpha_n}}.
\end{align*}
In other words, the $j$-th position in the multi-index tells how many derivatives are being taken with respect to the $j$-th variable.
For me, on the other hand, a multi-index $I$ can be any length, and is made up of integers from $1$ to $n.$ If $I = (i_1,\dots,i_k)$, then 
\begin{align*}
x^I &= (x_{i_1})\cdot \dots \cdot (x_{i_k}),\\
\partial_I &= \frac{\partial^k}{\partial x_{i_1}\dots \partial x_{i_k}}.
\end{align*}
In other words, the $j$-th position tells which variable the $j$-th derivative is taken with respect to.
These different conventions result in different ways of bookkeeping for all the terms in the Taylor polynomial. In Evans's notation, when you take a particular mixed $k$-th derivative, it only shows up once in the sum, but it shows up with a coefficient of the form $\alpha! = (\alpha_1)!\dots(\alpha_n)!$, just from the formula for taking derivatives of a polynomial. On the other hand, in my notation, each $k$-th derivative shows up multiple times, one for each possible ordering of the indices, and each of these gets a coefficient depending on which variables are differentiated how many times. But magically, when you add the terms all up, the overall coefficient turns out to be exactly $k!$. (You don't have to believe in magic -- just follow the proof in my book! It's nothing but integration by parts.)
