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As other correspondents have pointed out, this is essentially a theta function. You ask if you can write it in any other way. You can replace the infinite series by an infinite product :-) One gets $$ …

To give a solution from scratch, start with the identity $$\sum_{s=-\infty}^\infty\frac1{(x+s)^2}=\frac{\pi^2}{\sin^2\pi x}.$$ We want to prove that this extends to an identity $$\sum_{s=-\infty}^\inf …

It follows from Lebesgue's characterization of Riemann integrable functions as bounded functions continuous outside a set of Lebesgue measure zero.

You want to show that $$\int_{-\pi}^\pi|\log|1+e^{-it}||dt$$ is finite. Now $$|1+e^{-it}|=|e^{it/2}+e^{-it/2}|=2\cos(t/2)$$ so your integral is $$\int_{-\pi}^\pi|\log|2\cos(t/2)||dt =2\int_0^\pi|\log| …

It's a divergent infinite product. You might as well ask for the sum of $$\sum_{n=1}^\infty\frac{x}{n\pi}.$$ You can "cure" the divergence by multipliying each term by a suitable factor, so for instan …

By Taylor's theorem $$\phi(t+h)=\phi(t)+h\phi'(t)+h^2\phi''(t+u(h,t)h)/2$$ where $0\le u(h,t)\le1$. So $$\phi_h(t)=\phi'(t)+h\phi''(t+u(h,t)h)/2.$$ As $\phi''$ is in $C^\infty_c$ it's pretty clear tha …

It is a standard result that each closed subset of $\mathbb{R}^n$ is a zero set of some smooth function on $\mathbb{R}^n$. One proves this using smooth bump functions and partitions of unity.

See Conway's base 13 function.

There are proofs in textbooks such as Lang's Real and Functional Analysis and Conlon's Differentiable Manifolds but they do use some Banach space theory (not an awful lot). I have also seen in one wel …

This is discussed briefly as a generalization of Shapiro's cyclic sum inequality by J. Michael Steele in his book The Cauchy-Schwarz Master Class. He remarks that (1.) holds for $n\ge25$ and refers to …

It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the …