Einstein Tensor Notation: Addition inside a function Main Question
Can I represent addition of multi-dimensional variables in this linear function in Einstein Summation Convention?
$$
f(\mathbf{x} + \mathbf{v})
$$
This didn't seem right $f(x_{\mu}+v_{\mu})$ and nor did $f(x+v)_{\mu}$
More Information...
I am doing an unrelated question on multi-dimensional Taylor Series and I wanted to test my knowledge of compact tensor notation.
My full expression is
$$
   f = f(x_1,\cdots,x_N) = f(\mathbf{r})
   \quad\text{and}\quad
   \mathbf{v} = \left(h_1,\cdots,h_N\right)
   \\
   f( \mathbf{r} + \mathbf{v} ) = 
   \mathbf{v} \cdot \nabla f( \mathbf{r} )
   + \sum^{N}_{n=2} \frac{1}{n!} \left( \mathbf{v} \cdot \nabla \right)^n
   f(\mathbf{r}) + R_N ( \mathbf{r}, \mathbf{v})
$$
I am looking to compactly represent this in tensor notation as I am sure it can be done.
 A: With tensor notation I assume you just mean Einstein's summations convention, i.e. the convention where repeated indices are summed over all the coordinates instead of having an explicity sum: $a_ib_i \equiv \sum_{i=1}^n a_i b_i$.
The notation $f({\bf x})$ is just a shorthand for $f(x_1,x_2,\ldots,x_n)$, i.e. to tell the reader that $f$ takes points in $\mathbb{R}^n$ as it's argument. The point $(x_1,\ldots,x_n)$ can be written as the sum $x_\mu e^\mu$ where $e^\mu$ are basis-vectors in $\mathbb{R}^n$, for example $e^\mu=(0,\ldots,0,1,0,\ldots,0)$, however writing $f(x_\mu e^\mu)$ is not standard and can be confusing. You are better off using one of the two standard ways mentioned above with $f({\bf x})$ being the most compact one.
The summation convention is often very useful when doing calculations, the final result of such calculations often has a more clear and compact formulation using vector-calculus expressions like $\nabla$, dot-product, $\times$, etc. The notation should be used wisely. Personally I think the formula you have looks better as written than expressed in the summation convention (i.e. ${\bf v}\cdot \nabla f$ is more clear and compact than $v_i\frac{\partial f}{\partial x_i}$) and also note that the sum $\sum_{n=2}^N$ cannot be removed even when using the summation convention (without additional definitions).
A: After discussion with a few course mates we came to the conclusion that it should generally be left as the following,
$$
f(\mathbf{x} + \mathbf{v}) = \cdots
$$
with index notation on other relevant terms.
On the topic of taylor series for tensors there is a really interesting treatment of Taylor's Theorem in Tensor Calculus presented in this 1929 paper. 
