Math 104 homework 4 determine the convergence or divergence

Question

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 (a_n) be a sequence where 0 ≤ a_n < 1 for n ∈ N. Prove that if _n=1∑^∞a_n converges, then so do _n=1∑^∞a_n² and n=1∑∞a_n/(1 - a_n). Are the converse statements true?

3. 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 apart from at finitely many n, then ∑u_n or ∑v_n either both converge or both diverge.

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

7. Find a sequence (a_n) such that _n=1∑^2Na_n and _n=1∑^2N+1a_n both converge as N → ∞, but ∑a_n 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 d_n that a stack of bricks from 1 to n can overhang a brick (n+1). Does ∑d_n converge or diverge?