Is $\sum_{n=1}^{\infty }\:\left(\cos\left(\frac{1}{n}\right)-\cos\left(\frac{1}{n+1}\right)\right)$ the same as $\sum_{n=1}^{\infty }\cos\left(\frac{1}{n}\right)-\sum_{n=1}^{\infty }\cos\left(\frac{1}{n+1}\right)$?
I know both separately diverge based on the divergence test. But when subtracted they equal 0, does that mean divergence it is inconclusive?
The series can be viewed as a limit,
$$\lim_{N\to \infty}\sum_{n=1}^N(\cos(1/n)-\cos(1/(n+1))=\lim_{N\to \infty}(\cos(1)-\cos(1/(N+1))=\cos(1)-1 \,\, .$$