Determine whether the series is convergent or divergent. If convergent, find the sum; if divergent, enter div. \[ \sum_{n=1}^{\infty} \arctan (17 n) \] Answer:

We first observe that the general term of the series is \[ a_n = \arctan(17n). \] To determine if the series \[ \sum_{n=1}^{\infty} \arctan(17n) \] converges or diverges, we check the behavior of the term \(a_n\) as \(n\) approaches infinity. As \(n \to \infty\), we have \[ \lim_{n \to \infty} 17n = \infty. \] Recall that \[ \lim_{x \to \infty} \arctan(x) = \frac{\pi}{2}. \] Thus, \[ \lim_{n \to \infty} \arctan(17n) = \frac{\pi}{2}. \] Since the limit of the terms is not \(0\), one of the necessary conditions for the convergence of an infinite series is not met. Therefore, by the Divergence Test (also known as the \(n\)th term test for divergence), \[ \text{if } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges.} \] Since \[ \lim_{n \to \infty} \arctan(17n) = \frac{\pi}{2} \neq 0, \] the series \[ \sum_{n=1}^{\infty} \arctan(17n) \] diverges. Answer: div.