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Note 10 - 03/04/2023

TABLE OF CONTENTS

I. Basic limits

  1. $\lim_{n \to \infty} \dfrac{1}{n^\alpha} = 0$ if $\alpha > 0$
  2. Let $L=\lim_{n \to \infty} r^n$, then:
    • $L=0$ if $\vert r \vert <1$
    • $L=1$ if $r=1$
    • $L=\infty$ if $r>1$
    • $L$ does not exist if $r \leq -1$

  3. Squeeze theorem

    If $a_n \leq b_n \leq c_n, \forall n \in \mathbb{N}$ and $\lim a_n = \lim c_n = L$ then $\lim b_n = L$

  4. $\lim a_{2n}=\lim a_{2n+1}=L \Rightarrow \lim a_n=L$ $\lim a_{2n} \neq \lim a_{2n+1} \Rightarrow \lim a_n$ does not exist.
  5. Weierstrass’s theorem Let $\{a_n\}$ be a monotonic bounded sequence then $\{a_n\}$ converges.