Math 104: Homework 4 (due February 18)

1. Ross exercise 11.10

2. Ross exercise 12.10

3. Ross exercise 14.13

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

(a) ∑_{n} \frac{n^{3}}{2^{n}}, (b) ∑_{n} \sqrt{n + 1} − \sqrt{n}, (c) ∑_{n} \frac{1}{\sqrt{n!}}, (d) ∑_{n} 2^{−3n+(−1)^{n}} , (e) ∑_{n} \frac{n!}{n^{n}}.

5. Let (a_{n}) be a sequence where 0 ≤ a_{n} < 1 for n ∈ . Prove that if ∑ _{n=1}^{∞} a_{n} converges, then so do ∑ _{n=1}^{∞} a_{n}^{2} and ∑ _{n=1}^{∞} a_{n}/(1 − a_{n}). Are the converse statements true?

6. Let (u_{n}) and (v_{n}) be sequences of positive real numbers for 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}^{2N} a_{n} and ∑ _{n=1}^{2N+1} a_{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{matrix}[figure: d_{1}; 2; 1] & [figure: d_{1}; d_{2}; 3; 2; 1] \\ (a) & (b) \end{matrix}

Begin by considering diagram (a): what is the maximum distance d_{1} that brick 1 can overhang brick 2 without falling? Now, by considering combined center of mass of bricks 1 and 2, find the distance d_{2} 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?