Common Methods to Prove Limit of Infinite Sequence Exists – Analysis

analysislimitssequences-and-series

I would like to know what are common methods can be used to show that an infinite sequence converges. From what I know so far,

  1. If a sequence is bounded and monotonic increasing/decreasing then it converges.
  2. Using definition of limit.
  3. Another method that I saw online, is to assume the sequence approaches a limit $L$, then solve for $L$, but I'm not totally convinced that this approach is correct. For example, the Fibonacci ratio sequence, to prove the limit of $$\displaystyle\lim_{n\to\infty} \dfrac{a_{n+1}}{a_n}$$ exists, they claim that:
    $$1 + \dfrac{1}{L} = L$$
    proof for 3

So I wonder could anyone could share me some most commonly used method for proving the limit of an infinite sequence exists that I'm not aware of? Any suggestion or ideas would be appreciated.

Best Answer

Here is a theorem tells you if a sequence converges to zero or diverges to infinity,

Theorem: If ${a_n}$ be a sequence such that $\lim_{n\to \infty} \frac{a_{n+1}}{a_n}= a\,,$ then

1) if $|a|<1$, then $\lim_{n\to \infty}a_n =0 \,,$

2) if $ a>1$, then $\lim_{n\to \infty}|a_n| =\infty \,.$

See here for applications.

Related Question