The degree of a morphism of schemes

In this paper1 in section 2, a method to write the elements of the coset of G/H is provided for GL(4), but I am interested in $GL^+$(2).

My matrix representation of $mathfrak{gl}(2)$ is

$$
begin{bmatrix}
a+x& -b+y\
b+y & a-x
end{bmatrix}
$$

My matrix representation of $mathfrak{so}(2)$ is

$$
begin{bmatrix}
0& -b\
b & 0
end{bmatrix}
$$

Reading the paper, it states that the element g of G can be decomposed as g=$gamma$h, where

$$
gamma = exp left( begin{bmatrix}
a+x& +y\
y & a-x
end{bmatrix} right) in G/H
$$

and where

$$
h = exp left( begin{bmatrix}
0& -b\
b & 0
end{bmatrix} right) in H
$$

Is this correct, or no?

I am suspicious of the argument, because to me

$$
exp left( begin{bmatrix}
a+x& y\
y & a-x
end{bmatrix} right)exp left( begin{bmatrix}
0& -b\
b & 0
end{bmatrix} right) neq exp left( begin{bmatrix}
a+x& -b+y\
b+y & a-x
end{bmatrix} right)
$$

Thus, $gamma h$ does not appear to realize all elements of $G$. Or do we not care about some missing elements for cosets?

This exercise specifically requires that we use the root test to determine whether the series converges or not.

All I’ve done so far is get the sequence in this form:
$$sqrt[n] frac{n^{n+frac{1}{n}}}{(n+frac{1}{n})^{n}} = sqrt frac{n^{(1+n^{-2})n}}{(n+frac{1}{n})^n}=frac{n^{1+n^{-2}}}{n+frac{1}{n}} = frac{n^{frac{n^2+1}{n^{2}}}}{n^{2}+1}$$

But, I’m not even sure if I’m on the right track here. Any guidance is appreciated.

Edit: I showed my effort. I didn’t tell anyone to solve it for me. Not sure why the downvotes..

I would like to verify the following identity but I don’t know how mathematics says that it is equal to numerically.
$$prod _{k=0}^{infty } sqrt{frac{phi ^{2^{-k-1}} left(phi ^{2^{-k}}+1right)}{phi ^{2^{1-k}}+1}}=frac{sqrt{2}}{sqrt[4]{5}}$$

Let $P(k, n) : exists b_{1}, b_{2}, …, b_{n} in mathbb{N}, (forall i in mathbb{Z^+}, 1 leq i leq k implies b_{i} leq i) land(k = sum limits _{i=1}^n b_{i} cdot i!)$ where $k in mathbb{N}$ and $n in mathbb{Z^+}$.

How to show with induction $forall k in mathbb{N}, forall n in mathbb{N^+}, k < (n+1)! implies P(k,n)$

Any suggestions and hints would be appreciated.

In this note from Keith Conrad, he explains an interesting application of Strassmann’s theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $a_0 = a_1 = 1$ and $a_{m} = 2a_{m-1} – 3a_{m-2}$ satisfies $lim_{mto infty} |a_m| = infty$. It seems somewhat easy to prove at first glance (and the actual behavior of $|a_m|$ is exponential), one needs to deal with possible cancellation. The general term is given by
$$
a_m = frac{1}{2} ((1 + sqrt{-2})^{m} + (1 – sqrt{-2})^{m}) = sqrt{3}^m cos (m alpha)
$$
where $alpha = arctan(sqrt{2})$, and if $malpha$ is sufficiently close to the odd multiple of $pi / 2$, then $cos(malpha)$ could be very close to zero. So one may need to show that $cos(malpha)$ can’t be exponentially small with respect to $m$ in some sense, or use $p$-adic method as in Conrad’s note.

What I thought is that once we know $lim_{mtoinfty}|a_m| = infty$, this would tell us that $alpha / pi$ can’t be very close to rational numbers, and may tell something about irrationality measure of $alpha / pi$. I tried some but didn’t get anything useful at this moment. Is it possible to deduce some information on the irrationality measure of $alpha /pi$ using divergence, at least finiteness or infiniteness?

Can someone explain how I should approach solving the integral displayed in the image (4.10)? The integral is solving for the average velocity orthogonal to region in the second picture. We are only interested in the velocity going from the reactant region to the product region. This is part of a derivation for transition state theory and I am confused on how to approach solving it.