Vtyjkyuk

In my notes I have that

$$sum_m a_m delta_nm=a_1delta_n1+a_2delta_n2+a_3delta_n3+cdots=a_ntagA$$

Is this really correct?

I thought that for the Kronecker delta the first index must match the summation index. For this reason, I thought that the expression $(mathrmA)$ should written as

$$sum_colorbluem a_m delta_colorbluemn=a_1delta_1n+a_2delta_2n+a_3delta_3n+cdots=a_ntagB$$

Just to make my argument clear, I have made the color of the indices match.

After looking at this page on the Kronecker delta I know that it is okay to write
$$sum_ja_jdelta_ij=a_itag1$$
or
$$sum_ia_idelta_ij=a_jtag2$$

Expression $(1)$ matches $(mathrmA)$ and expression $(2)$ matches $(mathrmB)$

So does this mean that switching the order of the indices of the Kronecker delta (within a summation) has no effect on the result (the RHS), or am I missing something?

The Kronecker symbol $delta_ij$ is used to say $delta_ii=1$ and $delta_ij=0$ if $i neq j$.

This is independent of any summation. Kronecker symbol can be used in general. That being said, the evaluation of formula $(A)$ is perfectly correct as the only term that is not vanishing is when index $m$ is equal to $n$.

By its definition, $delta_i,j=delta_j,i,$ [one can drop the commas as you have] one can switch the indices anywhere, in a sum or otherwise.