TITLE: What are the applications of sequence of functions?
QUESTION [7 upvotes]: This might be a "silly" question, but before starting my studies, I need a motivation.
In Analysis books, there are the subjects such as "sequence of functions, uniform convergence etc." which deals with basically the sequence of functions, but up to now [I'm a 2. year physics & mathematics student], I haven't seen any real application of the concept of sequence of functions. I mean, for example, I have seen lots of application of sequences in defining continuity, compactness etc. (our instructor does the whole analysis based on sequences), but this is not the case for the sequence of function. 
Therefore, my question is that what are the applications of the concept of "sequence of function" in both mathematics and physics ?

REPLY [3 votes]: As you are interested in physics, it is worth mentioning that in quantum mechanics, the wavefunction of a particle can be expanded as an infinite sum of eigenfunctions of the Hamiltonian operator via the Spectral Theorem (or an integral if the particle is unbound.) So, for example, if our Hamiltonian operator is $\hat{H}$, our eigenvalues are $E_1,E_2,\ldots$, and our eigenfunctions are $\psi_1,\psi_2,\ldots$ (that is, $\hat{H}(\psi_n) = E_n\psi_n$ for every $n \in \Bbb N$) then any wavefunction can be expanded as an infinite sum of these eigenfunctions. That is, for any wavefunction $\phi$, there exists $(c_n)$ such tht
$$
\phi = \sum_{k=1}^\infty c_k\psi_k, \tag{1}
$$
which means that the sequence of functions
$$
\phi_N = \sum_{k=1}^N c_k \psi_k \text{ converges to } \phi.
$$
By studying convergent sequences of functions, we learn important questions about when you can manipulate infinite series of functions term-by-term. Provided you can apply the Hamiltonian term-by-term, then
$$
\hat{H}\phi = \sum_{k=1}^\infty c_k \hat{H}\psi_k = \sum_{k=1}^\infty c_k E_k\psi_k.
$$
With a little more elbow grease, you can use this idea to reduce the full time-dependent Schrödinger equation so solving for all the eigenfunctions $(\psi_n)$ and expanding your initial conditions in the form of (1), upon which the time-dependent solution is 
$$
\phi(t) = \sum_{k=1}^\infty c_ke^{-iEt/\hbar} \psi_k.
$$
All of this only works because you can apply operators term by term to (1), which relies on theorems you will prove when studying series of functions.