For each series determine whether they converge or diverge

Question

Sample midterm 1 questions-

1. Prove that 1 + √(1 + √2) is irrational.

2. Consider the following series defined for n ∈ N:

∑ 8ⁿ/(n!)²,            ∑(-1)ⁿ/√(n² + n).

For each series, determine whether they converge or diverge. If you make use of any of the theorems for determining series properties, you should state which one you use.

3. (a) Let S and T be non-empty bounded subsets of R. Prove that sup S ∪ T = max{sup S, sup T} and sup S ∩ T ≤ min{sup S, sup T}.

(b) Extend part (a) to the cases where S and T are not bounded.

(c) Give an example where sup S ∩ T < min{sup S, sup T}.

4. Suppose that (s_n) is a convergent sequence and (t_n) is a sequence that diverges to ∞. Prove that lim_n→∞s_n + t_n = ∞.

5. (a) Let (s_n) and (t_n) be two sequences defined for n ∈ N. Prove that lim sup s_n + lim sup t_n ≥ lim sup(s_n + t_n).

(b) Construct an example where (lim sup s_n) · (lim sup t_n) ≠ lim sup(s_nt_n).

6. Let (a_n) and (b_n) are sequences defined for n ∈ N. Suppose that a_n → a and b_n → b as n → ∞ for some a, b ∈ R. If a_n ≤ b_n for all n, show that a ≤ b.

7. Consider the two sets

A = (0, 1] ∪ [4, ∞),           B = {1/2n : : n ∈ N}

For each set, determine its maximum and minimum if they exist. For each set, determine its supremum and infimum. Detailed proofs are not required, but you should justify your answers.

8. Consider the following series, defined for n ∈ N:

∑6ⁿ/nⁿ,                  ∑1/n + 1/2.

For each series, determine whether it converges or diverges. If you make use of any of the theorems for determining series properties, you should state which one you use.

9. Let S be a non-empty bounded subset of R. Define T = {|x|: x ∈ S} to be the set of all absolute values of elements in S. Prove that sup T = max{sup S, - inf S}.

10. Let (s_n) and (t_n) be two sequences defined for n ∈ N. Suppose lim s_n = ∞, and lim sup t_n < 0. Prove that lim s_nt_n = -∞.