Graduate Power series in quantum mechanics

[gremory]

TL;DR

Power series to solving the infinite square well

Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for $\psi(x)$ wouldn't give me the quantised energies. My solution was
$$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$, with $k = \frac{\sqrt{2mE}}{\hbar}$. I want to know what's the catch when doing this because we need the boundary conditions to quantize the energies and with this solution i don't see how i would get that.

 

[anuttarasammyak]

\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))=(\cos(x) + \sin(x)) \sum^{\infty}_{n=0} k^{2n}=\psi(x,k)

so something seems wrong. Where are the edges of infinite well ?