Summations and integrals with no upper limits

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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