MAT3, Mat-Øk 3, Mat-Tek3 -- Efterår 2014

Analyse 1

Onsdag den 17. september 2014,  kl. 08:15
Sted: Rum NJV14, 4-117.

Dagens program

08:15-10:15

Repetition og forelæsning: monotone følger, Cauchy følger, fuldstændigheden af de reelle tal.

10:30--12:00

Exercises: read examples 3.4 and 3.5 on page 190 in [Lay] (chapter 4, 'Sequences'). Also, exercise 8, 9 and 10 on page 195.
Hint for exercise 8: prove the identity $(s_{n+2}-s_{n+1})(s_{n+2}+s_{n+1})=4 (s_{n+1}-s_{n})$ and show by induction that the sequence is increasing if $s_2>s_1$ and is decreasing if $s_2< s_1$.

Hint for exercise 9: if $k=\sqrt{x}$, show that the sequence is constant. Now assume that $k\neq \sqrt{x}$ and show that $s_{n+1}-\sqrt{x}= \frac{(s_n-\sqrt{x})^2}{2s_n}>0$ for all $n\geq 1$. Also, show that $s_{n+1}< s_n$ starting from $n\geq 2$.

Literatur: afsnit 4.3 og 4.4 fra [Lay] (kapitel 4, 'Sequences').