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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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