  We demonstrate that certain classes of Schl\" omilch-like infinite series and
series that include generalized hypergeometric functions can be calculated in
closed form starting from a simple quantum model of a particle trapped inside
an infinite potential well and using principles of quantum mechanics. We
provide a general framework based on the Hilbert space eigenproblem that can be
applied to different exactly solvable quantum models. Obtaining series from
normalization conditions in well-defined quantum problems secures their
convergence.
