Convergence of series ${\sum_{n\ge1} \sqrt{n}\ln(\cosh(1/n))}$

Question

The series is $${\sum_{n\ge1} \sqrt{n}\ln(\cosh(1/n))}$$

I´ve tried with the quotient criterion and with the limit criterion but the critiria does not decide. I know that this series converges but I can´t find an upper bound for the sequence.

Answer 1

$$ \text{cosh}\left(\frac{1}{n}\right)\underset{(+\infty)}{=}1+\frac{1}{2n^2}+o\left(\frac{1}{n^2}\right) $$ Hence $$ \ln\left(\text{cosh}\left(\frac{1}{n}\right)\right)\underset{(+\infty)}{=}\ln\left(1+\frac{1}{2n^2}+o\left(\frac{1}{n^2}\right)\right)\underset{(+\infty)}{=}\frac{1}{2n^2}+o\left(\frac{1}{n^2}\right) $$ You finally have the following asymptotic behaviour

$$ \sqrt{n}\ln\left(\text{cosh}\left(\frac{1}{n}\right)\right)\underset{(+\infty)}{\sim}\frac{1}{2n^{3/2}}$$

What do you know about $\displaystyle \sum_{n \geq 1}\frac{1}{n^{3/2}}$ ?

Answer 2

Observe that when $n\to +\infty,$

$$\ln \left(\cosh \left(\frac 1n\right)\right)=\ln \left(1+\cosh \left(\frac 1n\right)-1\right) $$ $$\sim \cosh \left(\frac 1n\right)-1$$ $$= 2\sinh^2 \left(\frac {1}{2n}\right)\sim \frac {2}{4n^2}\sim \frac {1}{2n^2}$$

thus $$u_n=\sqrt {n}\ln \left(\cosh \left(\frac 1n\right)\right)\sim \frac {1}{2n^{\frac 32}} $$

and $\sum u_n $ converges.

Answer 3

For any $x\geq 0$ we have $\log\cosh x\leq \frac{x^2}{2}$, hence your series is convergent and bounded by $\frac{1}{2}\zeta\left(\frac{3}{2}\right)$.
Actually $$ \log\cosh\frac{1}{n} = \sum_{m\geq 0}\left(1+\frac{4}{\pi^2 n^2 (2m+1)^2}\right)=\sum_{m\geq 0}\sum_{h\geq 1}\frac{(-1)^{h+1} 4^h}{h \pi^{2h}n^{2h}(2m+1)^{2h}} $$ $$ \log\cosh\frac{1}{n} =\sum_{h\geq 1}\frac{(-1)^{h+1}(4^h-1)\zeta(2h)}{h\pi^{2h} n^{2h}}$$

$$ \sum_{n\geq 1}\sqrt{n}\log\cosh\frac{1}{n} =\sum_{h\geq 1}\frac{(-1)^{h+1}(4^h-1)\zeta(2h)\zeta(2h-1/2)}{h\pi^{2h}}$$ hence the original series can be bounded by $\frac{1}{2}\zeta\left(\frac{3}{2}\right)-\frac{1}{12}\zeta\left(\frac{7}{2}\right)+\frac{1}{45}\zeta\left(\frac{11}{2}\right)\leq\color{red}{\sqrt{5}-1}.$

Answer 4

By Taylor's formula with an integral remainder, we have
$$ \log \cosh x = \frac{x^2}{2} - \int_0^x \frac{4 \mathrm{e}^{2t} (\mathrm{e}^{2t} - 1)}{(\mathrm{e}^{2t} + 1)^3}(x - t)^2 \, \mathrm{d}t \le \frac{x^2}{2} $$
for all $x \ge 0$. Therefore,
$$ 0 < \sum_{n = 1}^\infty \sqrt{n} \log \cosh \left( \frac{1}{n} \right) \le \sum_{n = 1}^\infty \sqrt{n} \cdot \frac{1}{2n^2} = \frac{1}{2} \sum_{n = 1}^\infty \frac{1}{n^{3/2}} < \infty, $$
which shows that the series converges.