Files Course He NOTE: SHOW ALL WORK! UNSUPPORTED ANSWERS WILL NOT BE GIVEN CREDIT. 1) Use the Limit Comparison Test to establish the convergence or divergence of the series which is the sum from n=1 to infinity of the terms an= (2n2+3n)/[(5+n5)1/2] (10 points 2) Which do you think is more powerful, the Direct Comparison Test or the Limit Comparison Test? Explain your thoughts. (5 points) 3) Prove that you can use the integral test on the series which is the sum from n=1 to infinity of the terms an=In(n). Then use the test to establish the convergence or divergence of the series. (10 points) 4) Use the Ratio Test to discuss the convergence or divergence of the series which is the sum from n=1 to infinity of the terms an=[(-1)"] [n3]/3" (10 points) 5) Use the Alternating Series Theorem to prove that the series which is the sum from n=0 to infinity of an=(-1)"! converges. (10 points) 6) Do you consider that absolute or conditional convergence is the stronger, more useful form of convergence? Explain. (5 points) 7) Determine the interval of convergence of the power series which is the sum from n=0 to infinity of the terms an= [n(x+2)~]/(3 +1) (10 points 8) Are there power series which diverge everywhere? If so, exhibit one. If not explain why it cannot happen. (5 points) 9) Find the first three nonzero terms in the Maclaurin series of f(x)= tan x. (10 points) 10) If f(x)=T, + R, where T, is the nth Taylor Polynomial and R, is the remainder, what must happen to R, as n goes to infinity for the Taylor Series of f to actually equal f? (5 points) 11) (Extra Credit) Do some research! Find an example of a function which is continuous at x=0, infinitely differentiable at x=0, but whose Maclaurin series does not actually represent f in any interval around the origin. (5 points) 12) Find an equation of the ellipse with foci (0,2) and (0,-2) and vertices (0,3) and (0,-3) (10 points) 13) What is the distinguishing difference between the definition of an ellipse and the definition of an hyperbola? (the definition, not the equation) (5 points) MacBook Pro