In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, we have: $$a_{j+2}\approx\frac{2}{j}\,a_j.$$ Up to this point I totally agree (one just takes the limit).

However, the subsequent derivation of solution $a_{j}$ and $h(\xi)$ I attached from the Griffiths' textbook are very confusing.

• How did it go from $a_{j+2}\approx\frac{2}{j}a_j$ to the solution of $a_{j}$?
• Also, how do the second and third approximations work in $h(\xi)$?

My questions are mainly mathematical. I very much hope someone can provide a derivation or refer a link where these questions may already be answered.