Numerical analysis Homework Help

\text { (7) Suppose that } \lim _{n \rightarrow \infty} \sqrt{n} a_{n} \geq 1 \text { . Test the following series for convergence, } \sum_{n=1}^{\infty} \frac{a_{n}+\frac{1}{\sqrt{n}}}{n^{2} a_{n}^{2}}

Prove the following statements. [Each part is worth 5pt (a) Let {rn} be a converging sequence in a metric space X and let x € X be its limit. Use the definition of compactness to show that the set {x}U{xn} is compact. (b) Show that a subset of a metric space X is closed if and only if its intersection with every compact subset of X is closed.

(6) Let {an} be the sequence defined by a_{n+1}=\sqrt{2 a_{n}+3}, a_{1}=2 Show that {an} is convergent and find the limit.

\text { (1) If } A, B \text { are bounded sets of real numbers and } A \cap B \neq \varnothing \text { , show that } \inf (A \cap B) \geq \max \{\inf A, \inf B\}

Q6. Let (X, d) and (Y, p) be metric spaces. Show that the following two definitions of lower hemicontinuity are equivalent. [Each direction is worth 10pt]

\text { 2. Calculate } \lim \left(\sqrt{n^{2}+n}-n\right) \text {. }

6. Eigenvalue estimates [2+2+2pts] Gerschgorin's second theorem states that if the union of k Gerschgorin discs is disjoint from the other n - k discs, it must contain exactly k eigenvalues.Now let for some z E C. Here i is the imaginary unit. (a) Sketch the first two Gerschgorin discs for A. (b) Suppose we know that at least two of the three eigenvalues are equal. Using Gerschgorin's theorems, what can we conclude about the value of z? (Find the largest subset of C that you know z cannot be in) (c) Suppose we know that all eigenvalues are equal. What can we conclude about z?