(a) Start with the identity for the difference of tangents:

\(\tan(r+1) - \tan r = \frac{\sin(r+1)\cos r - \cos(r+1)\sin r}{\cos(r+1)\cos r}\)

This simplifies to:

\(\frac{\sin 1}{\cos(r+1)\cos r}\)

Thus, \(\tan(r+1) - \tan r = \frac{\sin 1}{\cos(r+1)\cos r}\).

(b) Using the method of differences, we have:

\(\sum_{r=1}^{n} u_r = \sum_{r=1}^{n} \frac{1}{\cos(r+1)\cos r}\)

Recognizing the telescoping nature:

\(\sum_{r=1}^{n} \left( \tan(r+1) - \tan r \right) = \tan(n+1) - \tan 1\)

Thus, \(\sum_{r=1}^{n} u_r = \frac{\tan(n+1) - \tan 1}{\sin 1}\).

(c) As \(n \to \infty\), \(\tan(n+1)\) oscillates, so the series \(u_1 + u_2 + u_3 + \ldots\) does not converge.