Intereting Posts

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One divided by infinity is not zero?
Proof of Cartesian product intersection
$\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I – A)}^{ – 1}}} \right\|}$
A question on a quotient of Alexandroff's double segment space
What is the simplest way to show that $\cos(r \pi)$ is irrational if $r$ is rational and $r \in (0,1/2)\setminus\{1/3\}$?
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Are isomorphic the following two links?
Sum of closed convex set and unit ball in normed space
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Induction proof (bitstring length)
Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.
Under what conditions does $(\frac{3}{p})(\frac{-1}{p})=1?$ Two ways, different results.

Given series $\sum_{n=1}^{\infty}\frac {\cos(n^2)}n$.

It is easy to prove it does not converge absolutely.

I need to prove that it converges сonditionally.

- Existence of a power series converging non-uniformly to a continuous function
- Power Series proofs
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- Proof for convergence of a given progression $a_n := n^n / n!$
- Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in $?
- Equicontinuity and uniform convergence 2

I thought about using Dirichlet’s test because $1/n$ series is monotone and $\lim_{n\rightarrow\infty} 1/n = 0$.

So the thing i need to prove is $\left|\sum_{1}^k \cos(n^2)\right| < M,\ \forall k$.

If there was $\cos(n)$ instead of $n^2$, it would be easy to prove this statement by multiplication and division by $\cos(0.5)$ and then using some trigonometric formula so that $\left|\sum_{1}^k \cos(n^2)\right| = \left|\frac{cos(0.5)-cos(n-0.5)}{\cos(0.5)}\right| < 2$ or something like that. But this approach seems to be impossible for $\cos(n^2)$.

Maybe there is a way to prove it using the fact that $\int \cos(x^2) = \sqrt{2/\pi}$?

Or any simpler way?

- Where is $f(x) := \sum_{n=1}^\infty \frac{\langle nx\rangle}{n^2+n}$ discontinuous?
- The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent
- Do these series converge to the von Mangoldt function?
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- Series $\sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n-1})^{\alpha}$ converge or diverge?
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- A theorem about Cesàro mean, related to Stolz-Cesàro theorem
- Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?
- Finding sum of the series $\sum_{r=1}^{n}\frac{1}{(r)(r+d)(r+2d)(r+3d)}$
- Find $\lim_{n \to \infty} \sqrt{n!}$.

It is sufficient to show that

$$\left|\sum_{n=0}^{N}\cos(n^2)\right|\leq C\sqrt{N}\log N\tag{1}$$

to ensure convergence by partial summation. Consider that:

$$\left\|\sum_{n=1}^{N}e^{in^2}\right\|^2 = \left(\sum_{n=1}^{N}e^{in^2}\right)\cdot\left(\sum_{n=1}^{N}e^{-in^2}\right)=N+\sum_{d=1}^{N-1}\sum_{r=1}^{N-d}2\cos(2dr+d^2),\tag{2}$$

and that:

$$(2)\ll \sum_{d=1}^{N-1}\min\left(N-d,\left\|\frac{d}{\pi}\right\|^{-1}\right)\tag{3}$$

(where $\|x\|$ denotes the distance of $x$ from the closest integer) by the usual arguments about simple exponential sums. If now we take $\frac{a}{q}$ as a good rational approximation of $\frac{1}{\pi}$, $\left|\frac{a}{q}-\frac{1}{\pi}\right|<\frac{1}{3Nq}$, it is not difficult to see that:

$$ (3)\ll \sum_{\substack{d=1\\q\nmid d}}^{N}\left\|\frac{a d}{q}\right\|^{-1}\ll(N+q)\log q\ll N\log N,\tag{4}$$

hence $(1)$ holds and the series $\sum_{n=1}^{+\infty}\frac{\cos n^2}{n}$ converges.

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- Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$
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- Why can't calculus be done on the rational numbers?
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- Image of measurable set under continuous inverse function is always measurable?