TITLE: What does this double sided arrow $\longleftrightarrow$ mean?
QUESTION [19 upvotes]: What is $\longleftrightarrow$ used for in mathematics? I know about $\iff$ being used for "If and only if". Are they the same thing? I was watching a YouTube video that said:
$$\sum^{\infty}_{n=1} {1\over n^x} \longleftrightarrow \int^{\infty}_{1} {1\over t^x} dt$$
The teacher mentions convergence/divergence, but I was confused when the notation came up.

REPLY [8 votes]: I understand it, in this context, as a sign corresponding to a "loose equivalence" between the convergence of both terms, under specific conditions that are not  fully mentioned. The presenter writes that a sum diverges/converges if the corresponding integral  diverges/converges. I would translate it as: "LHS property is (somehow) strongly related to the RHS one".
It is not an "if and only if", indeed this is not true in general in that case.
The $\leftrightarrow$ symbol appears after the Maclaurin–Cauchy integral test for  convergence (the so-called Cauchy integral theorem is quite different). The standard test works under the following conditions:

$f$ is continuous, defined on $[n_0, +\infty [$ for some integer $n_0$,
$f$ is monotone and decreasing.

Then the infinite series $\sum_{n=n_0}^\infty f(n)$
converges to a finite limit if and only if ($\Leftrightarrow$) the improper integral
$\int_N^\infty f(x)\,dx$ is finite. And if the integral diverges, then the series diverges as well. Here, the test works for the $p$-series, as $t \to \frac{1}{t^x}$ is continuous decreasing for $x >0$, and the convergence of the series depends on $x> 1$ or not.
As mentioned in comments, many mathematical symbols have several interpretations (e.g. bijection or logical biconditional).