Sigma question: is it legal to write something like this? Is this mathematical syntax correct?
$$\sum_{n+1}^m\sin(n-2)
$$
As you see, the starting value is $n+1$ instead of being just purely one variable.
 A: If there's any doubt about what the index of summation is, then specify it explicitly.  If you write about the sum of terms called $\sin(n-2)$, then commonplace conventions make the reader think $n$ goes from something to something.  But you've used $n$ as one of the bounds, meaning $n$ stays put while some other variable goes from $n+1$ to $m$, and what that other variable, the index, is called (is it $i$? is it $k$?) you don't say.  If you write
$$
\sum_{k=n+1}^m \sin(n-2),
$$
then that's
$$
\sin(n-2)+\sin(n-2)+\sin(n-2)+\cdots+\sin(n-2)
$$
and all terms are identical, and there are $m-n$ of them, so the sum is $(m-n)\sin(n-2)$.  If you meant anything other than that, then don't use this notation.
A: You have 
$$\sum_{n+1}^m\sin(n-2)$$
What is the running index here? Apparently $\,n\,$ , but from what number does it begin running? Perhaps it should be $\,n=1\,$ in the summatory's lower limit?
As it stands, the expression makes not much sense.
A: I would say that your notation is not good. The reason is that it isn't clear what the index of summation is. From how it is written it looks like $m$ and $n$ might both be constants. But then you only have the variable $n$ after the summation sign, so one would think that $n$ is what is "changing" in the summation. But if you want the sum to start at $n+1$, then you should write something like (as mentioned in the comments and the other answer): 
$$
\sum_{i = n+1}^m \sin(i-2).
$$
What this means is the sum
$$
\sin(n+1-2) + \sin(n+2-2) + \dots +\sin(m-1-2) + \sin(m-2).
$$
You could IMO get away with writing this same sum as
$$
\sum_{n+1}^m \sin(i-2).
$$
