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Too long for a comment: Here's a little intuitive tip: What do $~\dfrac{\sin t}t~$ and $~\dfrac{\cos t}{t^2}~$ both have in

common? They are even functions. So, if you notice various series or integrals whose summand

or integrand belongs to this category having a nice closed form, that should not surprise you. For

instance, $~\displaystyle\int_{-\infty}^\infty\frac{\sin x}x~dx=\pi,~$ or $~\displaystyle\int_{-\infty}^\infty\frac{\cos x}{1+x^2}~dx=\frac\pi e,~$ or $~\displaystyle\sum_{n=-\infty}^\infty'\frac1{n^{2k}}=a_k~\pi^{2k},~$ where the

apostrophe represents the omission of the divergent term corresponding to $n=0$, and $a_k\in\mathbb Q$.

Obviously, if one were to sum or integrate odd functions over this entire interval, then the result

would be $0$ for integrals, and either $0$ or $f(0)$ for sums, since the values on $(-\infty,0)$ would cancel

those on $(0,\infty)$. So, in this sense, if one were to define odd $\zeta$ values as $\zeta(2k+1)=\displaystyle\sum_{n=-\infty}^\infty'\frac1{n^{2k+1}}$

then they would indeed possess a very beautiful closed form, namely $0$. Indeed, $~\displaystyle\int_0^\infty\frac{\sin x}{1+x^2}~dx~$

also lacks a known closed form, as does $~\displaystyle\sum_{n=1}^\infty\frac{\sin nx}{n^2}.~$ Please do not misunderstand me, there are

exceptions to every rule, and one might indeed find counter-examples of both kinds, but usually

they are trivial $($e.g., the odd integrand whose primitive can be expressed in closed form, and then

evaluated at the extremities of the integration interval, or, in the case of $~\displaystyle\sum_{n=1}^\infty\frac{\cos nx}n,~$ the famous

Mercator series for the natural logarithm; not to mention a whole infinity of even functions whose

summation or definite integral simply does not possess a closed form, for the trivial reason that the

overwhelming majority of functions simply do not have one, and those that do are the exception

rather than the rule$)$.