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Should $\sum_{i=0}^n a + b$ be interpreted as $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$

I often see the expression $\sum_{i=0}^n a + b$ in books and wonder whether we take the sum $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$. Often I can see which one from context, but is there a general rule ?

Example is: $\sum^{n-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2}$. I misinterpreted this, because I assumed $\sum^{n-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2} = \sum^{n-1}_{j=0} ((-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2})$ but it was actually $(\sum^{n-1}_{j=0} (-1)^j x^{2j}) + (-1)^n \frac {x^{2n}} {1+x^2}$.

Shuzheng
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  • I think it is generally meant that $\sum_{i=0}^n a+b = \sum_{i=0}^n (a+b)$. With one exception: if the expression $b$ doesn't depend on $i$ then we have $\sum_{i=0}^n a+b =(\sum_{i=0}^n a)+b$. – Thomas Produit Mar 05 '14 at 18:22
  • It would be clearer if you wrote $\sum_{i=0}^na_i+b$ or $\sum_{i=0}^n(a_i+b_i)$. Now it seems that $\sum_{i=0}^na+b=(n+1)a+b$. – Cm7F7Bb Mar 05 '14 at 18:34

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There is no general rule. This is vague, and it is advisable to always include parenthesis. What is clear to you, might be vague to someone else. Also as you say, context can almost always be used to interpret though.

Guy
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  • As an example, I misinterpreted $\sum^{n-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2}$. I thought $(-1)^n \frac {x^{2n}} {1+x^2}$ was part of $\sum$, but it was not. – Shuzheng Mar 05 '14 at 18:45
  • In general if the "summing variable", in this case $j$ is not part of the term, that term is not in the $\sum$. In general though, there are plenty of times I write stuff like $n=\sum_1^n 1$ though, where it is useful. – Guy Mar 05 '14 at 18:47