I often see sum-notation wirtten in one of the following ways:
AND:
Is there any difference between the two, or is it just a matter of style? thanks.
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2I think it's just formatting. For MathJax, for instance, you get $\sum_{i=1}^{100}i$ in line, but $\displaystyle {\sum_{i=1}^{100}i}$ if it is offset (or, as here, marked as "display style"). – lulu May 09 '18 at 10:03
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I think the second is a typesetting error. I find it hard to read. It's interestion to note in MathJax $\sum_{i=1}^{100} i$ is rendered as $\sum_{i=1}^{100} i$ but $\sum\limits_{i=1}^{100} i$ is rendered as $\sum\limits_{i=1}^{100} i$, the latter is easier to read but harder to type. – fleablood May 09 '18 at 20:49
2 Answers
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There is no difference in concept.
I like the first one better because of its format.
Mohammad Riazi-Kermani
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Well, like for many mathematical symbols, sometimes the two forms are used equivalently, other times the different layout is useful to help and distinguish different definitions of Sum.
While the first layout is almost generally reserved for the "ordinary" sum, the second is used by many people (including me) to indicate the sum in the meaning of Antidifference, computed for $i$ ranging from $1$ to $100$.
That is
$$
f(x) = \Delta F(x) = F(x + 1) - F(x)\quad \Rightarrow \quad F(x) = \Delta ^{\, - 1} f(x) = \sum\nolimits_{\,x} {f(x)}
$$
which for integral upper and lower limits returns almost the same value as the ordinary sum
$$
\sum\nolimits_{\,i = 1\;}^{\,100} {f(i)} = F(100) - F(1) = \sum\limits_{i = 1}^{99} {f(i)}
$$
But with the Antidifference interpretation, we can give a meaning also to a sum having the upper bound less than the lower one $$ \sum\nolimits_{\,i = 100\;}^{\;1} { = - f(i)} \sum\nolimits_{\,i = 1\;}^{\,100} {f(i)} $$ and, if $F(x)$ is defined also for real or complex values of the argument, then the sum has a meaning also for real/complex bounds $$ \sum\nolimits_{\,i = 1\;}^{\,y} {f(i)} = F(y) - F(1)\quad $$ so quite a different concept.
G Cab
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