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