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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$$)$$.

 answered Aug 27 '15 at 19:43 Lucian 42.2k11 gold badge6464 silver badges134134 bronze badges