2. Use the Comparison Test or Limit Comparison Test to determine whether each series is convergent or divergent. Carefully explain your justification for using each test.

(a) Use the table to determine a midpoint Riemann Sum with n = (b) Interpret your answer in the context of that situation. What are you estimating with this Riemann Sum?

(Squeeze theorem, sandwich theorem, squeeze lemma) (a) Let (an), (bn), (cn) be real consequences with

(a) By means of induction according to n, show that an 2 2 holds for all n 21. (b) Show that the sequence is monotonically decreasing. (c) Show that the sequence converges and determine its limit value.

2. Prove that a metric space X is discrete if and only if every function on X to an arbitrary second metric space is continuous.

4. The Reimann zeta-function is defined by and is used in number theory to study the distribution of primes. (a) For values of a does converge? (i.e What is the domain of (?) Explain your reasoning. (b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4

A 4 (a) Let (an)nen and (ba)nen be two sequences in R such that the series [n=1, and [mb2 are absolutely convergent. Show that then also the series