Reference no: EM131081523
Math 104: Homework 4-
1. Determine the convergence or divergence of each of the following series defined for n ∈ N:
(a) ∑nn3/2n,
(b) ∑n√(n + 1) - √n,
(c) ∑n1/√n!,
(d) ∑n2-3n+(-1)^n,
(e) ∑nn!/nn.
2. Let (an) be a sequence where 0 ≤ an < 1 for n ∈ N. Prove that if n=1∑∞an converges, then so do n=1∑∞an2 and n=1∑∞an/(1 - an). Are the converse statements true?
3. Let (un) and (vn) be sequences of positive real numbers for n ∈ N. For each of the following statements, either prove it or provide a counterexample.
(a) If (un) and (vn) are equal apart from at finitely many n, then ∑un or ∑vn either both converge or both diverge.
(b) If (un) and (vn) are equal at infinitely many n, then ∑un or ∑vn either both converge or both diverge.
(c) If (un/vn) → 1 as n → ∞, then ∑un and ∑vn both converge or both diverge.
(d) If un - vn → 0, then ∑un and ∑vn both converge or both diverge.
(e) If (un+1/un) > k > 1 for infinitely many n, then ∑un diverges.
7. Find a sequence (an) such that n=1∑2Nan and n=1∑2N+1an both converge as N → ∞, but ∑an is divergent.
8. Optional for the enthusiasts. Consider an infinite number of bricks of unit length, made from a uniform material.
Begin by considering diagram (a): what is the maximum distance d1 that brick 1 can overhang brick 2 without falling? Now, by considering combined center of mass of bricks 1 and 2, find the distance d2 that can they can overhang brick 3. Now determine the maximum distance dn that a stack of bricks from 1 to n can overhang a brick (n+1). Does ∑dn converge or diverge?
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