Are there nice functions for which $\sum\limits_{n\geq 1} f(n) = \int\limits_{\mathbb{R}_+}f(t)dt$? What can we say about the class of functions for which $$\sum\limits_{n\geq 1} f(n) = \int\limits_{\mathbb{R}_+}f(t)dt$$
Are there any good examples of such functions?
Edit: You may prefer different summation/integration bounds like $n \geq 0$ or $t \geq 1$, doesn't really matter.
 A: By the Abel-Plana formula, if $f$ is an entire even function whose modulus does not grow too fast on the right half-plane,
$$ \sum_{n\geq 0} f(n) = \frac{f(0)}{2}+\int_{0}^{+\infty}f(x)\,dx. \tag{1}$$
Folklore: Growth conditions are a bit tricky, though. You might be induced to think that, by this principle,
$$ \frac{1}{2}\left(\frac{\pi^{1/4}}{\Gamma(3/4)}+1\right)=\sum_{n\geq 0} e^{-\pi n^2} \stackrel{\color{red}?}{=} 1 $$
but the difference between the LHS and the RHS, albeit small, is not zero. It is $\approx 0.04322$.
On the other hand $(1)$ works like a charm for computing series like
$$ \sum_{n\geq 1}\left(\frac{\sin n}{n}\right)^k\quad\text{or}\quad \sum_{n\geq 1}\left(\frac{1-\cos n}{n^2}\right)^k$$
or for series involving values of Bessel functions.
A: If $f(x) = \lfloor{x\rfloor}$, an example would be
$$\int_{0}^{n}f(x)dx = \sum_{k=0}^{n-1}f(k)\ =\frac{n\left(n-1\right)}{2},$$
or equivalently,
$$\int_{0}^{n}\lfloor{x\rfloor}dx = \sum_{k=0}^{n-1}k\ =\frac{n\left(n-1\right)}{2}.$$
