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In the Sigma-notation the index variable bound to the Sigma sign is said to be unrelated to the appearances of the variable in the summand. For example, in the sum

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

the subscript $k$ in $a_k$ is said to be not linked to the $k$ in $\Sigma_{k=1}^n$. Consequently it can be replaced by another letter and it will not change the meaning of the sum. For example, the following is equivalent

$\Sigma_{k=1}^n a_m$

My question is why is the index variable bound to the Sigma sign is unrelated. It seems counter-intuitive. I thought the index variable linked to the Sigma sign is related because the values it takes will be used in the summand.

  • You thought xorrectly, and $\sum_{k=1}^n a_k \ne \sum_{k=1}^n a_m$ except sometimes by accident. – André Nicolas Jan 28 '14 at 16:11
  • I don't know where you have seen this, but it's wrong. The $k$ in the Sigma sign is related to the $k$ used in $a_{k}$. The second formula you wrote is equivalent to $n*a_{m}$ since in regard to the variable $k$ used in the sigma sign, $a_{m}$ is constant. Your last sentence is correct. – Traklon Jan 28 '14 at 16:11
  • I heard this from one of my statistic lecturers. I have also seen it in a book called Concrete Mathematics (by Donald Knuth et al) on page 22, 2nd paragraph. But to be clear the second sum was added by me to explain what I thought my lecturer meant and what I inferred from the book. –  Jan 28 '14 at 16:30
  • What you can say is that $\Sigma_{k=1}^n a_k=\Sigma_{m=1}^n a_m$. Choosing $k$ or $m$ or any other letter as index variable is irrelevant. – Américo Tavares Jan 29 '14 at 12:27
  • This is the exact quote from Professor Knuth's book: "The index variable $k$ is said to be bound to the $\Sigma$ sign in $\Sigma_{k=1}^n a_k$, because the $k$ in $a_k$ is unrelated to appearances of $k$ outside the Sigma-notation". To me that can lead to misinterpretation. That is, there is disconnect between the index variable bound to the Sigma and the one in the summand. –  Jan 30 '14 at 10:01

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This question has been answered in comments:

I don't know where you have seen this, but it's wrong. The $k$ in the Sigma sign is related to the $k$ used in $a_k$. The second formula you wrote is equivalent to $n∗a_m$ since in regard to the variable $k$ used in the sigma sign, $a_m$ is constant. Your last sentence is correct. – Traklon Jan 28 at 16:11

and

What you can say is that $\sum^n_{k=1}a_k=\sum^n_{m=1}a_m$. Choosing $k$ or $m$ or any other letter as index variable is irrelevant. – Américo Tavares Jan 29 at 12:27

Note that in

"The index variable $k$ is said to be bound to the $\sum$ sign in $\sum^n_{k=1}a_k$, because the $k$ in $a_k$ is unrelated to appearances of $k$ outside the Sigma-notation".

the key word if outside. For example, you might have $2\pi i+\sum_{i=1}^n a_i$. However, since using the same letter to denote both the summation variable and something else could cause confusion, it would be good form to avoid doing so.

Jessica B
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