Series expansion of a harmonic oscillator

[speedofdark8]

Homework Statement

Use a series expansion ψ=A₀x⁰+A₁x¹+A₂x²+... to determine the three lowest-order wave functions for a harmonic oscillator with spring constant k and mass m, and show that the engergies are the expected values.

Homework Equations

Series expansion given above

Time independent Schrodinger's equation

The Attempt at a Solution

I am not familiar with series expansion. Asking others in the class, we got as far as what is in the picture, where we took the function for a harmonic oscillator F=-kx and substituted it for U in Schrodinger's Equation.

 

[dikmikkel]

You would want to read this under the harmonic oscillator.
http://spacsun.rice.edu/~baring/phys516/phys516_2008_lect0916.pdf