One mistake you made is when you said $\lfloor\frac m2\rfloor=\lfloor\frac{m-1}{2}\rfloor +1$. This is only true when $m$ is even. I find it is much easier to split into cases based on whether $m$ is even or odd. If $m$ is even, so that $m=2k$ for some $k\in \mathbb N$, then $$ \begin{align} F_{m...

Solve the following differential equation: $\frac{d^2y}{dx^2}+a^2y=\cos ax.$ The solution given is as follows: The complementary function is $c_1\cos ax + c_2\sin ax$; the particular integral is $\frac{1}{D^2+a^2}(\cos ax)=\frac{1}{-a^2+a^2}\cos ax$; and thus the method fails. In this case, chan...

Suppose that $k$ is a field extension of field $f$. I don't understand one step in the proof of $[k:f]<\infty\implies |Gal(k/f)|<\infty.$ Proof: If $[k:f] < \infty$, then $k$ is finitely generated extension of $f$, i.e., $k= f(a_1, ..., a_k)$. I know that $k$ automorphisms are determined by actio...