Reference no: EM131081210
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 (sn) is a convergent sequence and (tn) is a sequence that diverges to ∞. Prove that limn→∞sn + tn = ∞.
5. (a) Let (sn) and (tn) be two sequences defined for n ∈ N. Prove that lim sup sn + lim sup tn ≥ lim sup(sn + tn).
(b) Construct an example where (lim sup sn) · (lim sup tn) ≠ lim sup(sntn).
6. Let (an) and (bn) are sequences defined for n ∈ N. Suppose that an → a and bn → b as n → ∞ for some a, b ∈ R. If an ≤ bn 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 (sn) and (tn) be two sequences defined for n ∈ N. Suppose lim sn = ∞, and lim sup tn < 0. Prove that lim sntn = -∞.
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