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If I had a summation such as:

$$\sum_{i=0}^\frac{n}{2} n+1 $$

Is there a way I could do it, or is it just undefined notation?

I thought a switch with the fractions like this would work:

$$\sum_{i=0}^n \frac{n+1}{2} $$

As in, double the terms but halve the value.

I came up with it while toying with Gauss' formula for the sum of the first n terms.

RobPratt
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Ampersands
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  • The summand does not depend on $i$ so it is just $(n+1)$ times the number of terms (be that $\frac n2+1$ or $\lfloor \frac n2\rfloor +1$, it's not clear what you intended.) – lulu Apr 10 '23 at 00:13
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    In general, we sometimes write $$\sum_{0\leq i\leq n/2},$$ or $$\sum_{i=0}^{\lfloor n/2\rfloor}$$ Sometimes, we just write it as $$\sum_{i=0}^{n/2}$$ to mean the same as the other, more correct notations. – Thomas Andrews Apr 10 '23 at 00:16
  • My three notations are still sims for integers $i.$ – Thomas Andrews Apr 10 '23 at 00:17
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    Depends on what you mean. First, the summand should be in terms of $i.$ It's not wrong to write: $$\sum_{i=0}^k(n+1),$$ but it just means $(n+1)+(n+1)+\cdots+(n+1),$ $k+1$ times. – Thomas Andrews Apr 10 '23 at 00:18

1 Answers1

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It really depends on what do you want to do.

Without any prior explanation given,

$$\sum_{i=0}^{n} a_i$$

just means that you add $1$ to the index each time, and add the corresponding $a_i$. You ignore anything for $i>n$.

Correspondingly, the most natural interpretation of

$$\sum_{i=0}^{5/2} a_i$$

(as an example) is still just $a_0 + a_1 + a_2$, since $3 > 5/2$. Most would clarify such edge cases, however, by taking the ceiling or the floor of the upper bound, i.e.

$$\sum_{i=0}^{\lfloor 5/2 \rfloor} a_i \qquad \sum_{i=0}^{\lceil 5/2 \rceil} a_i$$

depending on whichever fit their use case better.

If you want to add in increments of $1/2$ instead, as I feel this may be suggesting, you should either

  • Change the indices inside the sum to accommodate this. For instance, if I have a set of numbers $a_0,a_{1/2},a_1,a_{3/2},a_2,a_{5/2},\cdots$ and so on, I might realize that the numbers take the form $k/2$ for some $k \ge 0$. So then I might have $$ \sum_{i=0}^5 a_{i/2} = a_0 + a_{1/2} + a_1 + a_{3/2} + a_2 + a_{5/2} $$

  • Use a different notation. For finite indexing sets $S$ for sure, the notation $$ \sum_{i \in S} a_i $$ is unambiguous. So you might define, say, $$ S_n := \left\{ \frac k 2 \; \middle| \; k \in \{0,1,\cdots,n\} \right\} $$ and therefore $$ \sum_{i \in S_5} a_i = a_0 + a_{1/2} + a_1 + a_{3/2} + a_2 + a_{5/2} $$ Of course, you could tweak the set and indexing as you please, for whatever form is convenient. Just be wary of concerns for infinite and especially uncountable sets.

PrincessEev
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