DO 1. a. Explain why the statement is true: If ak converges, then lim ak = 0. k-+0o k=1 b. Explain why its converse is not necessarily true: If lim ak = 0, then E ak converges. k=14. Apply the Divergence Test. State Whether each series diverges or whether the test is inconclusive: 0 1 \"'3\" 1+3\" 03n2+1 21.: n+1 ling? \"'2 4n d'E74n2 73:1 7. a. Explain how we can conclude that f() = 2x2 - 1 is continuous when x 2 1 . b. Explain how we can conclude that f (x) = 2x2 - 1 is positive when x 2 1 . c. Use the derivative to show that f(x) = 2x2 - 1 is decreasing when x 2 1 . n d. Apply the Integral Test to show that E diverges. n=1 2n2 - 1 8. Check that the assumptions of the Integral Test are satisfied, then apply the test to determine 1 whether V3n -5 converges or diverges. n=210. A p-series is of the form E kp where p is a real constant. K=1 a. Use the Integral Test to show that this series converges for p = 1.5 and diverges for p = 0.5. b. What do we call this series when p = 1? Does it converge or diverge? c. For what values of p does the p-series converge or diverge