# Test the following series for convergence..

#### General

$$\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{n^{sinh(n)}}$$

#### chisigma

MHF Hall of Honor
Because is $$\displaystyle \sinh 1 = 1.1752011936438...$$ and the function $$\displaystyle \sinh (*)$$ is [strongly] increasing, for all n will be...

$$\displaystyle \frac{1}{n^{\sinh n}} < \frac{1}{n^{1.1}}$$ (1)

Now the series...

$$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{1.1}}$$ (2)

... converges so that...

Kind regards

$$\displaystyle \chi$$ $$\displaystyle \sigma$$