Series of $(\arctan(\cos n))^n$ I was thinking of a way to prove that the series $\begin{aligned}\sum_{n=0}^{\infty} \end{aligned} (\arctan(\cos n))^n$ converges (applying the WynnEpsilon method in NSum of Wolfram Mathematica I found to be about equal to 1.56021). I was thinking of a strategy as proposed here, but they immediately (and rightly) denied me. Now I do not know what to think. Some idea? Thank you!

The root test says the sum converges if

$$\limsup_{n\to\infty}|\arctan(\cos n)|<1$$

Since we know that

$$\sup_{n\ge0}|\arctan(\cos n)|=\arctan(\cos(0))=\arctan(1)=\frac\pi4$$

It follows that

$$\limsup_{n\to\infty}|\arctan(\cos n)|\le\frac\pi4<1$$

So the given series converges.