+87 1. Find all \(r\) for which the infinite geometric series \(2 + 6r + 18r^2 + 54r^3 + \dotsb\)
is defined. Enter all possible values of \(r\) as an interval.
2. Compute \(1 - 2 + 3 - 4 + \dots + 2005 - 2006 + 2007\)
3. Let \(f(x) = \frac{x^2}{x^2 - 1}\) Find the largest integer \(n\) so that \(f(2) \cdot f(3) \cdot f(4) \cdots f(n-1) \cdot f(n) < 1.98\)
1. You want the geometric series to converge, so -1 < r < 1.
2. 1 - 2 + 3 - 4 + ... + 2005 - 2006 + 2007 = (1 - 2) + (3 - 4) + ... + (2005 - 2006) + 2007 = 1002.
3. The product will telescope, and you are left with n/(n - 1), so you want n/(n - 1) < 1.98. The largest n that satisfies this is n = 201.
