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I reading on Sums and I am reading about the difference between using a generalized Sigma notation and the delimited form. Ok, I understand that the generalized form is more expressive.

But I found the following example confusing to me:
It says:

In the following we can change the index variable from $k$ to $k+1$ and easily do the substitution: $$ \sum_{1 \leqslant k \leqslant n} a_k = \sum_{1 \leqslant k+1 \leqslant n} a_{k+1}. $$

But with the delimited form we have:
$$ \sum_{k=1}^n a_k = \sum_{k=0}^{n-1} a_{k+1}; $$

and is harder to make a mistake.

My question is the following:
In the generalized form the index is expressed to be $k+1$ in the notation so we have $k+1=1$, $k+1=2$.

But why is the comparison done like this?
I mean shouldn't we have in the right hand of the delimited form $k+1=1$ instead?
I.e.

$$ \sum_{k=1}^n a_k = \sum_{{\Large \mathbf{k+1=1}}}^n a_{k+1}. $$

Or is it not allowed to use a complex index in the delimited form?

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The notation

$$ \sum_{k=1}^{n} a_k$$

is short form of the following

$$ \sum_{k=1}^{k=n} a_k$$

So, by changing index variable from $k$ to $k + 1$ we will get

$$ \sum_{k+1=1}^{k+1=n} a_{k+1} = \sum_{k=0}^{k=n-1} a_{k+1} = \sum_{k=0}^{n-1} a_{k+1} $$

Note: In general this is just a notation (or mutual agreement). There is a lot of forms of summing. You can use any as long as other people can understand you.

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A complex index in delimited form is never used, the main reason being that this is highly problematic for non-linear variable changes if you think of the delimited form as running through values 1 to n and the non-delimited form as a system of conditions that happen to be inequalities in your case.

If the variable change is linear, your proposed notation will not lead to problems and will certainly be understood, but I would not advice to use it.

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