(ii) Let f(x) = 1+x- x' + x' and g(x) =1-r+x? - r. Prove that there exists a CER such that f (g(c)) + g (f(c)) = 0. 3. (a) [3 marks] Let {a, } C R. Consider the series _ a,. Draw a fully labeled Venn diagram to illustrate the relationship between absolute convergence, conditional convergence and divergence for the series _ an. (b) [4 marks] Determine whether the series E -2n 5n n=12 n+1 absolutely converges, conditionally converges, or diverges. Make sure to fully justify your claim. 5. [13 marks] Find the radius of convergence and the interval of convergence for the power series (-1)n (2n - 1 )20 (X - 1)n. 6. [8 marks ] Let a E R. Let {o,} C R. Suppose that the sequence {a, } converges to a. Provide a complete and accurate e - N proof that {a, } converges. 9. [5 marks] Let {a, } be a sequence defined recursively by 01 = 3, an+1 =- 4 - an - for n 2 1. Suppose that 0