# 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.

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5
It may even be legal, depends on the judge. But it is a really bad idea to use $n$ as the (presumed) summation index, and also as a component of one of the ends. If I am reading your intent correctly, I would write something like $\displaystyle\sum_{i=n+1}^m \sin(i-2)$. – André Nicolas Sep 23 '12 at 18:31

## 3 Answers

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.

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what i want to mean is for the sum start at n+1 – user1561559 Sep 23 '12 at 18:29
@user1561559 yes sure. But summand is always prametrized. In your case it must be $n$ as otherwise doesnt make sense. If $m>n+1$, you can define such but I am not sure if it makes sense... – Seyhmus Güngören Sep 23 '12 at 18:31

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@user1561559 Then you probably want $$\sum_{i=n+1}^m\sin(i-2)$$ unless the expression you are going for is equal to $(m-n-1)\sin(n-2)$. – Alex Becker Sep 23 '12 at 18:33

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.

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IMO, the index of summation should be specified explicitly even when there can't really be any doubt – really, it's just one symbol! The range over which that variable is summed, which may be rather more awkward to write, may be well be left out if it's clear from the context like in $\langle a, b \rangle = \sum_i a_i\!\cdot\!b_i$. (Which, of course, you might write simply $a_ib_i$, following Einstein convention...) – leftaroundabout Sep 23 '12 at 21:58

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 $$\sum_{n+1}^m \sin(i-2).$$

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