Determine the convergence or divergence of each of the

Question

Math 104: Homework 4-

1. Determine the convergence or divergence of each of the following series defined for n ∈ N:

(a) ∑_n n³/2n'

(b) ∑_n√(n + 1) - √n,

(c) ∑_n 1/√(n!)'

(d) ∑_n2^-3n+(-1)n,

(e) ∑_n n!/nⁿ.

2. Let (u_n) and (v_n) be sequences of positive real numbers for n ∈ N. For each of the following statements, either prove it or provide a counterexample.

(a) If (u_n) and (v_n) are equal except at finitely many n, then ∑u_n and ∑v_n either both converge or both diverge.

(b) If (u_n) and (v_n) are equal at infinitely many n, then ∑u_n and ∑v_n either both converge or both diverge.

(c) If (u_n/v_n) → 1 as n → ∞, then ∑u_n and ∑v_n both converge or both diverge.

(d) If u_n - v_n → 0, then ∑u_n and ∑v_n both converge or both diverge.

(e) If (u_n+1/u_n) > k > 1 for infinitely many n, then ∑u_n diverges.

3. Find a sequence (an) such that _n=1∑^2Naⁿ and _n=1∑^2N+1a_n both converge as N → ∞, but ∑a_n is divergent.

4. 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 d_n that a stack of bricks from 1 to n can overhang a brick (n+1). Does ∑d_n converge or diverge?