(a) Consider \(w_{r+1} = (r+1)(r+2)\ldots(r+10)\) and \(w_r = r(r+1)(r+2)\ldots(r+9)\).

Then, \(w_{r+1} - w_r = (r+1)(r+2)\ldots(r+10) - r(r+1)(r+2)\ldots(r+9)\).

Factor out \((r+1)(r+2)\ldots(r+9)\):

\(= (r+1)(r+2)\ldots(r+9)((r+10) - r) = 10(r+1)(r+2)\ldots(r+9)\).

(b) The sum \(\sum_{r=1}^{n} u_r = \frac{1}{10}((w_2 - w_1) + (w_3 - w_2) + \ldots + (w_{n+1} - w_n))\).

This telescopes to \(\frac{1}{10}(w_{n+1} - w_1)\).

\(w_{n+1} = (n+1)(n+2)\ldots(n+10)\) and \(w_1 = 1 \cdot 2 \cdot \ldots \cdot 10 = 10!\).

Thus, \(\sum_{r=1}^{n} u_r = \frac{1}{10}((n+1)(n+2)\ldots(n+10) - 10!)\).

(c) The series \(v_1 + v_2 + \ldots + v_n = x^{w_2} - x^{w_1} + x^{w_3} - x^{w_2} + \ldots + x^{w_{n+1}} - x^{w_n}\).

This telescopes to \(x^{w_{n+1}} - x^{w_1}\).

For convergence as \(n \to \infty\), \(|x| < 1\) is required.

Thus, for \(-1 < x < 1\), the sum to infinity is \(-x^{w_1} = -x^{10!}\).

For \(x = \pm 1\), the sum to infinity is 0.