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.

Questions 9-10. use the table of educational expenditures to answer the questions. k-12 education expenditures population indiana $9.97 billion 6.6 million illinois $28.54 billion 12.9 million kentucky $6.58 billion 4.4 million michigan $16.85 billion 9.9 million ohio $20.23 billion 11.6million 9. if the amount of k-12 education expenses in indiana were proportional to those in ohio, what would have been the expected value of k-12 expenditures in indiana? (give the answer in billions rounded to the nearest hundredth) 10. use the value of k-12 education expenditures in michigan as a proportion of the population in michigan to estimate what the value of k-12 education expenses would be in illinois. a) find the expected value of k-12 expenditures in illinois if illinois were proportional to michigan. (give the answer in billions rounded to the nearest hundredth) b) how does this compare to the actual value for k-12 expenditures in illinois? (lower, higher, same? )