Why don't we indicate the variable to summed as we do for integrals?

Question

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so:

$$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes deeper into single- and especially multivariable calculus, where one discovers that it does't just signify which variable to integrate.

But is there no valid reason to write, for example, the sum $1+1/4+1/9+\dots$ in this fashion:

$$\sum_{1}^{\infty} \frac{dn}{n^2}$$

Instead of the usual:

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

Has it ever been done?

Answer 1

The $dx$ or $dn$ parts are used in integrals to show which variable we're integration over and so we can abuse the notation.

For sums and products the variable we're summing over is written below the sum or product sign, there's no reason to write which variable we're summing over twice.

Answer 2

The $dx$ you see in an integral is cosmically related to the $\Delta x$ you see in a Riemann sum:

$$ \sum_{i=1}^n f(c_i^*) \Delta x_i $$

where $c_i^*$ is the sample point in the $i$-th interval, and $\Delta x_i$ is the width of the $i$-th interval. The $dx$ is often thought of as the "infinitely small" version of $\Delta x_i$. This is similar to the $dx$ in the denominator of $\frac d{dx}$, which stands for an infinitesimal $\Delta x$.

Answer 3

Well, that's just the way the notation works, but there are very good reasons for it.

We could have used something like

$$\int_{x=1}^{\infty} \frac{1}{x^2}$$

to keep it consistent, in a sense, but the meaning and placement of the differential $dx$ is a bit more subtle than just an index of a summation, and makes perfect sense.

The notation $\int f(x) dx$ means summing ("$\int$") the product "$f(x) \, dx$", not summing "$f(x)$" over "$dx$". The original reason for this is based on the notion of infinitesimals, but you can associate $dx$ to the term $\Delta x$ on a Riemann sum, which also gets multiplied by the integrand.

This usage is conceptually different than the way indices are used in summations.

But in principle, it's just notation.

Answer 4

The "d$x$" is best regarded as a mnemonic symbol, and it reminds the reader (though somewhat misleading for novices) how an integral is carried out. In Russell's term, "d$x$" is called an incomplete symbol, which has no meaning itself when escaping from a given context.

I do not see apparent reasons keeping an author away from writing $\sum_{1}^{\infty}\frac{1}{n^{2}}dn,$ and indeed I write so when I wish to emphasize the similarity between the discrete and the continuous. Nonetheless, I think in discrete case d$n$ is simply = 1.