Hey GeeRaCeR.
I don't know what you have covered, but there is a result known as the Euler-Mclaurin formula which links sums, integrals, and error bounds together. Take a look at the wiki in relation to the basel problem (and also note that the basel problem can be written in terms of an integral with a fourier series decomposition):
Euler?Maclaurin formula - Wikipedia, the free encyclopedia