This is a special fact about $\mathrm{SU}(2)$. One looks at the action of diagonal matrices $\mathrm{diag}(u,1/u)$ on the representation. An irreducible representaion breaks up into "weight spaces" where this matrix multiplies the vector by $u^j$. The representation $V_j$ splits into one-dimensional weight spaces of weights $-j$, $-(j-2)$, $-(j-4),\dots,j$. To count the number of irreps of a given type in a finite-dimensional representation one does a census of the weight spaces. If you do that for $V_i\otimes V_j$ you find that the stated irreps turn up just once each.

That's a misprint: I meant $x=4.5$. So $f_1(4.5)=0$, $f_2(4.5)=0$, $f_3(4.5)=0$, $f_4(4.5)=1$, $f_5(4.5)=0$, etc.

Take for example $n=4.5$. Then the sequence $(f_n(x))$ goes 0, 0, 0, 1, 0, 0, and it stays at 0 from then on, so it converges to 0. Exactly the same thing will happen for any $x$.

The (co)homology of $K(G,1)$ is well-known to equal the group (co)homology of $G$ with integer coefficients. I don't know what happens for $n>1$.

I find it rather odd calling a number a "constant". What precisely makes a number like $\pi$ a "constant"? Surely it's just an interesting number?

Your opinions are normative statements: "one should" and "it is better". It is naive to suppose that there is one best method that one should use to compute the matrix exponential.

Higham has a whole chapter (10) on the matrix exponential, most of which can be found on Google books.

Even a cursory examination of Nick Higham's book amazon.co.uk/Functions-Matrices-Computation-Nicholas-Higham/‌​dp/… will show that both these opinions on the evaluation of matrix exponentials are hopelessly naive.

The convergence of this general form is related to the irrationality measure of $\pi$, that is the infimum of exponents $k$ such that $|\pi-a/b|<1/b^k$ has only finitely many integer solutions. (For $|\sin n|$ to be small, $n$ must be close to an integer multiple $m\pi$ of $\pi$ and then $|\sin n|\sim m|\pi-n/m|$.) Results are known (see for instance planetmath.org/encyclopedia/IrrationalityMeasure.html) and these will yield explicit values of $a$, $b$ for which the series converges, but the proofs are delicate and don't yield the best expected result.