Section outline

  • A sequence

    {an}\{a_n\}

    {an​} converges to a real number

    LL

    L if:

    ∀ϵ>0,∃N∈N such that n>N⇒∣an−L∣<ϵ\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n > N \Rightarrow |a_n - L| < \epsilon

    We write

    lim⁡n→∞an=L\lim_{n \to \infty} a_n = L

    .
    If no such

    LL

    L exists, the sequence diverges.

    Basic limit theorems:

    • lim⁡(an±bn)=lim⁡an±lim⁡bn\lim (a_n \pm b_n) = \lim a_n \pm \lim b_n

    • lim⁡(anbn)=(lim⁡an)(lim⁡bn)\lim (a_n b_n) = (\lim a_n)(\lim b_n)

    • lim⁡(anbn)=lim⁡anlim⁡bn\lim \left( \frac{a_n}{b_n} \right) = \frac{\lim a_n}{\lim b_n}

      , provided

      lim⁡bn≠0\lim b_n \neq 0