Provide exact answers to the following, except when otherwise noted. (Recall that x = √ 2 is an exact solution to the equation x 2 = 2, while x = 1.414213562 is an approximate solution, and that you will not have access to calculators on exams.) 1. For what values of p does the series X[infinity] n=1 ln n np converge? 2. Show that if an ≥ 0 and X[infinity] n=1 an converges, that X[infinity] n=1 (an) 2 does too. 3. Show that if an ≥ 0 and X[infinity] n=1 an converges, that X[infinity] n=1 ln(1 + an) does too. 4. Show that if an ≥ 0 and X[infinity] n=1 an converges, that X[infinity] n=1 sin(an) does too.