It is quite a well known fact that:
$$\int_0^{+\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$$
also the value of related series is very similiar:
$$\sum_{n = 1}^{+\infty} \frac{\sin n}{n} = \frac{\pi – 1}{2}$$
Combining these two identities and using ${\rm sinc}$ function we get:
$$\int_{-\infty}^{+\infty} {\rm sinc}\, x \, dx = \sum_{n = -\infty}^{+\infty} {\rm sinc}\, n = \pi$$
What is more interesting is the fact that the equality:
$$\int_{-\infty}^{+\infty} {\rm sinc}^k\, x \, dx = \sum_{n = -\infty}^{+\infty} {\rm sinc}^k\, n$$
holds for $k = 1,2,\ldots, 6$. There are some other nice identities with ${\rm sinc}$ where sum equals integral but moving on to other functions we have e.g.:
$$\sum_{n = -\infty}^{+\infty} \binom{\alpha}{n} e^{int} = \int_{-\infty}^{+\infty} \binom{\alpha}{n} e^{itx} \, dx = (1+e^{it})^\alpha, \; \alpha >-1$$which is due to Pollard & Shisha.
And finally the identity which is related to the famous Sophomore's Dream:
$$\int_0^1 \frac{dx}{x^x} = \sum_{n = 1}^{+\infty} \frac{1}{n^n}$$
Unfortunately in this case the summation range is not even close to the interval of integration.
Do you know any other interesting identities which show that "sum = integral"?
Best Answer
Several papers are dedicated to the subject of integrals of functions that equal the sum of the same function, primarily for estimation purposes.
Boas and Pollard (1973) has some interesting sum-integral equalities:
$$\pi/\alpha=\sum_{n=-\infty}^\infty \frac{\sin^2 (c+n)\alpha}{(c+n)^2}=\int_{-\infty}^\infty \frac{\sin^2 (c+n)\alpha}{(c+n)^2}\, \text{d}n$$
$$\pi\operatorname{sgn} a=\sum_{n=-\infty}^\infty \frac{\sin (n+c)\alpha}{n+c}=\int_{-\infty}^\infty \frac{\sin (n+c)\alpha}{n+c}\, \text{d}n$$
It also gives several general formulae for functions that suffice:
$$\sum_{n=-\infty}^\infty f(n)=\int_{-\infty}^\infty f(n) \, \text{d}n$$
mainly with Fourier analysis.
This paper gives an equality with the Bessel J function:
$$\int_{-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}\, \text{d}t=\sum_{t=-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}$$
and some more references:
See also Surprising Sinc Sums and Integrals which has some other equalities. This paper also states that (paraphrasing)
If $G$ is of bounded variation on $[−\delta, \delta]$, vanishes outside $(−α, α)$, is Lebesgue integrable over $(−α, α)$ with $0 < α < 2\pi$ and has a Fourier transform of $g$, then
$$\sum_{n=-\infty}^\infty g(n)=\int_{-\infty}^\infty g(x)\, \text{d}x+\sqrt{\frac{\pi}{2}}(G(0-)+G(0+))$$
Ramanujan's second lost notebook contains some sums of functions that equal the integral of their functions (Chapter 14, entries 5(i), 5(ii), 16(i), 16(ii)).
If you want, even more references with examples are in the papers I have mentioned.