Homework 5. Due on October 23

1. Decide if the following series converges, give your reasoning:

(a) ∑ \frac{n^{2}−5n+3}{10n^{2}+5} (b) ∑ \frac{1}{n^{2}−3n+6} (c) ∑ \frac{sinn}{n^{2}} (d) ∑ \frac{1}{n lnn}

2. If ∑ x_{n} and ∑ y_{n} are convergent, prove that ∑(x_{n} + y_{n}) is convergent.

3. Can you give an example of a convergent series ∑ x_{n} and a divergent series ∑ y_{n} such that ∑(x_{n} + y_{n}) is convergent? Explain.

4. If ∑ a_{n} with a_{n} > 0 is convergent, then is ∑ a_{n}^{2} always convergent? Is ∑ \sqrt{a}_{n} always convergent? Either prove or give counter a counterexample in each case.

5. Suppose {a_{n}} is a decreasing sequence of strictly positive numbers. Prove that ∑ a_{n} converges if and only if ∑ 2^{n}a_{2^{n}} converges.

6. Use problem 5 to discuss the convergence or divergence of the p -series ∑ \frac{1}{n^{p}}.

7. Problem 9 on page 79 of Rudin. (You may treat z as a real number).

8. We say a sequence {x_{n}} of real numbers is contractive if there exists a constant C, 0 < C < 1 such that |x_{n+2} − x_{n+1}| ≤ C|x_{n+1} − x_{n}| for all n ∈ ℕ. Prove that a contractive sequnce is a Cauchy sequence.

9*. Decide the convergent or divergent of the series ∑ \frac{cosn}{n} and ∑ \frac{sinn}{n}.