Why can a perturbed wavefunction be written as a power series? In Griffith's Quantum Mechanics, Time-Independent Perturbation Theory, the author writes the solution to the perturbed hamiltonian ($H = H^0 + \lambda H'$) as a power series in the variable $\lambda$: 
$$\psi_n = \psi_n^0 + \lambda \psi_n^1 + \lambda^2 \psi_n^2 + \ldots$$
and similarly it's eigenvalues as:
$$E_n = E_n^0 + \lambda E_n^1 + \lambda^2 E_n^2 + \ldots$$
What are the mathematical and physical motivations for this expansion? 
 A: Perturbation theory typically leads to a divergent asymptotic expansion. Such an expansion is still useful in practice, yielding a good approximation when taking the first few terms of the expansion. A simple argument due to Dyson can often be invoked to demonstrate this. The classic example is the quartic oscillator:
$$H = \frac{p^2}{2m} +\frac{m\omega^2}{2}x^2 + \lambda x^4$$
for $\lambda>0$. It's straightforward to compute the first few terms of the pertubative expansion in powers of $\lambda$, you'll get good approximations for small $\lambda$ by taking the first few terms of this expansion. But the physics of this problem will tell you to not expect a convergent series, because for negative $\lambda$ there are no bound states, so for that case there cannot be a convergent series, at least not to the correct answer. If we assume that the series diverges for negative $\lambda$, then because power series have a radius of convergence, it follows that this radius of convergence must be zero.
This is Dyson's change of sign argument against convergence, which is not mathematically rigorous as it ignores the possibility that the series might still converge to some finite value for negative $\lambda$. But this intuitive argument usually does correctly predict the divergence of the perturbation series.
The reason why the series looks like converging for small $\lambda$ if you take only a few terms, is because for small negative $\lambda$ the potential will only start to divergence to negative infinity very far away, the unperturbed wavefunction will be exponentially small there. So, from the point of view of the system itself it looks like there is no problem, the unperturbed wavefunction only has a significant amplitude at regions where the perturbation is small.
So, on physical grounds, you would expect that the series starts out just like a convergent series, but that the fact it is actually divergent will eventually manifest itself; the smaller $\lambda$ is chosen the later the terms will start to diverge. This is indeed the typical pattern that is observed for perturbation series.
A: The eigenvalues are analytic functions of $\lambda$ (under certain conditions).
Kato proves this for finite dimensions (formula (2.36) p. 81) and on p. 370 says "In this way the study of a finite system of eigenvalues ... has been reduced to that of a problem in a finite-dimensional space". (The link downloads the book from University of Edinburgh.)
