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"?