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Math Help - convergence

  1. #1
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    convergence

    If  a_n is a sequence such that  \frac{a_{n}-1}{a_{n}+1} \to 0 , then does  \lim_{n \to \infty} a_{n} exist? So in other words, does  a_n \to L ?

    Assume that  s_n = \frac{a_{n}-1}{a_{n}+1} . Let  \varepsilon > 0 . Then there exists an  N \in \mathbb{N} , such that whenever   n \geq N , then  |s_{n}| < \varepsilon .

    Now the problem is to express  a_{n} in terms of  s_{n} and use the definition of convergence to see if it converges or not?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello
    Quote Originally Posted by particlejohn View Post
    If  a_n is a sequence such that  \frac{a_{n}-1}{a_{n}+1} \to 0 , then does  \lim_{n \to \infty} a_{n} exist? So in other words, does  a_n \to L ?

    Assume that  s_n = \frac{a_{n}-1}{a_{n}+1} . Let  \varepsilon > 0 . Then there exists an  N \in \mathbb{N} , such that whenever   n \geq N , then  |s_{n}| < \varepsilon .

    Now the problem is to express  a_{n} in terms of  s_{n} and use the definition of convergence to see if it converges or not?
    If you have to use the definition of convergence, that's the idea. To help you, |s_n|<\varepsilon can be written -\varepsilon < 1-\frac{2}{a_n+1} < \varepsilon.

    If you don't have to use the definition of convergence, suppose that (a_n) diverges and see what happens to s_n=1-\frac{2}{a_n+1} which is supposed to tend to 0 as n\to\infty.
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