Problem with Free Index in Einstein Summation Notation From http://www.physics.ohio-state.edu/~ntg/263/handouts/tensor_intro.pdf:

Rules of Einstein Summation Convention — If an index appears (exactly) twice, then it is summed over and appears only on one side of an equation.
  A single index (called a free index) appears once on each side of the equation. So
  $A_{\LARGE{i}} = B_{\LARGE{i}}C_{\LARGE{i}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,\, \, \, \, \, \, \, \, \, \, \, \, \, (1)$ is INvalid.
  $A_{\LARGE{i}} = \epsilon_{\LARGE{ijk}}B_{\LARGE{i}}C_{\LARGE{j}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {(2)}$ is INvalid.

I understand (1) is invalid because there's 1 $i$ on the LHS but $2$ on the RHS. But I don't understand the rationale behind this rule? What's the problem?
$\sum_{i=1}^n A_i$ = $\sum_{i=1}^n B_iC_i $ is valid because it means $A_1 + ... + A_n = B_1C_1 + ... + B_nC_n $.
I understand (2) is invalid — On the LHS, when the summation is expanded in $i$, there's no $k$. However, on the RHS, when the 2 summations are expanded, $k$ is still there in the Levi-Civita tensor.
 A: Basically the point is that it is ambiguous to use an index as both a free and dummy index. 
$$ A_iB_{ii} \stackrel{\huge{?}}{=}  A_iB_{jj} = A_i \sum_j B_{jj}$$
or 
$$ A_iB_{ii} \stackrel{\huge{?}}{=} A_jB_{ji} = \left(\sum_j A_jB_j\right)_i ? $$
This is a problem, we can't have both. 
Added after Trapu's edit. So, to be clear, I will be non-einstein notation on the r.h.s. of the equations below: the point here is that $A_iB_{ii}$ cannot be interpreted meaningfully treating one pair of $i$ as dummies (summed over) and the other as free (not summed)
$$ A_iB_{ii} = \sum_{j=1}^n A_jB_{ji} = A_1B_{1i}+A_2B_{2i}+ \cdots +A_nB_{ni} \qquad (I.) $$
verses:
$$ A_iB_{ii} = \sum_{j=1}^n A_iB_{jj} = A_i\sum_{j=1}^n B_{jj} = A_i\left(B_{11}+B_{22}+ \cdots +B_{nn} \right) \qquad (II.) $$
Expressions (I.) and (II.) are two reasonable interpretations of $A_iB_{ii}$ if just one index is taken to be free. But, these are not equal.
For example, $B_{11}=1, B_{22}=-1, B_{12}=0=B_{21}$ and $A_1=1, A_2=1$,
$$ (I.) \qquad A_1B_{1i}+A_2B_{2i} = B_{1i}+B_{2i} = \begin{cases} 1 & i=1 \\ -1 & i=2 \end{cases} $$
verses
$$ (II.) \qquad A_i(B_{11}+B_{22}) = 0 = \begin{cases} 0 & i=1 \\ 0 & i=2 \end{cases} $$
As you can see these expressions do not agree. Therefore, we cannot use the same index for a dummy and a free index.
I would liken this problem to that I have with my calculus I students who insist they need not change the bounds in a u-substitution since they're just going to write it back in terms of $x$ at the end. However, if such a practice is made then some of the intermediate steps are wrong. We are left with the situation that what we write is insufficient to capture the precise mathemtical intent of the expression. This should be avoided since good notation ought to be unambiguous. Or, at a minimum the ambiguity should reflect a deeper mathematical structure as in the case of quotient spaces and the non-uniqueness of the representative. This is not that, this is just bad notation. It does lead to errors, trust me, I've made them.
