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I've seen expressions like:

$$\sum\limits_{i} f(x)$$

And $$ \int\limits_\mathbb{R}f(x)dx $$

What does it mean that they have no upper limits?

YoTengoUnLCD
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4 Answers4

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Let's give an example for summation: $$ \sum_k \binom{n}{k}(-1)^k \left( 1 - \frac{k}{n} \right)^n $$ Since $\binom{n}{k}$ is defined as being $0$ except when $0 \leq k \leq n$ this sum is the same as $$ \sum_{k=0}^n \binom{n}{k}(-1)^k \left( 1 - \frac{k}{n} \right)^n $$ and is neater to write.

By the way, as a softball follow-on question, show that the sum we are talking about is $$\frac{n!}{n^n}$$

As to integrals over $\Bbb{R}$ that just means an integral from $-\infty$ to $+\infty$. But I am disturbed by not seeing the $dx$ in the integrand.

Mark Fischler
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The first expression is just plain bad. It doesn't make sense because $f(x)$ has no obvious dependence on the index $i$.

For the second, we're just integrating along the whole real line $(-\infty, \infty)$. We often use the term improper to describe such integrals, because they may still be poorly defined, such as if $f(x)$ has a singularity (e.g. $f(x) = 1/x$). We can occasionally settle such issues by being more precise, such as by using the Cauchy principal value.

Christopher A. Wong
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The second is clear by itself. It means integrate through the entire $\Bbb R$. The first one need context to explain, usually the author wants to same sum all $i$ as mentioned about, or it is an infinite series.

MonkeyKing
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  • @By integrate through the entire $\mathbb{R}$ you mean $\int\limits_{-\infty}^{\infty}f(x)$? – YoTengoUnLCD Apr 30 '15 at 00:09
  • @YoTengoUnLCD Yup. – MonkeyKing Apr 30 '15 at 00:09
  • What if the lower limit is not $\mathbb{R}$? I've seen some expressions like that but can't recall any examples at the moment... – YoTengoUnLCD Apr 30 '15 at 00:13
  • It doesn't mean "lower limit". Sometimes people put the domain of integral at the bottom. It is convenient when the domain is not subset of $\Bbb R$, for example, a high dimentional ball, usually denoted as $B_r(P)$, where $r$ is the radius, $P$ is the center. – MonkeyKing Apr 30 '15 at 00:17
  • $\int_{\Omega} f(x) dx$ means " the integral of $f$ over the set $\Omega$" – Tryss Apr 30 '15 at 00:17
  • @Tryss and how would that be evaluated? Taking the upper limit as the "biggest" element in $\Omega$? – YoTengoUnLCD Apr 30 '15 at 00:18
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    Forget about the "upper limit". $\int_a^b f(x) dx$ is just a notation for "the integral of f over the set $[a,b]$" (modulo the question of the sign that is a little tricky) – Tryss Apr 30 '15 at 00:21
  • The reason you keep thinking of "upper limit" and "lower limit" is because of the fundamental theorem of calculus. It applies differently in general, and sometimes it is not useful. – MonkeyKing Apr 30 '15 at 00:23
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For the first:

When the index set is understood from context, it is often dropped, leaving only the index, as in ∑i i2. This will generally happen only if the index spans all possible values in some obvious range, and can be a mark of sloppiness in formal mathematical writing. Theoretical physicists adopt a still more lazy approach, and leave out the ∑i part entirely in certain special types of sums: this is known as the Einstein summation convention after the notoriously lazy physicist who proposed it.

http://www.cs.yale.edu/homes/aspnes/pinewiki/SummationNotation.html#Sums_without_explicit_bounds

Win Myo Htet
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