Mathematics - Mathematical Analysis Subsequences and Convergence

    Hello its me again drifter1. Today we continue with Mathematical Analysis getting into Subsequences and the Convergence of Sequences. I suggest you to check out my previous post about the Basics of Sequences here before getting into this post! So, without further do, let's get straight into it!

Subsequence:

    Subsequence of an sequence (an) is every sequence (bn) with generic term b(n) = a(kn), for every natural number n and with k(n) being a strictly increasing sequence of natural numbers. That's why we write a subsequence if (an) as (akn) for the specific sequence (kn).

Example:

    If we take only the odd terms a(1), a(3), ..., a(2n-1), ... of a sequence (an), then end up with a subsequence (bn) with generic term b(n) = a(2n-1), for every natural number n and for a sequence (kn) with generic term k(n) = 2n-1. We could do the same with even terms and end up with c(n) = a(2n) and so l(n) = 2n. The "sum" of those two subsequences equals the given sequence. So, this means that we can split a sequence into many subsequences and the "sum" of those will give us our first sequence.

Subsequence Boundary and Monotony:

  • If a sequence (an) is bounded (upper and lower) then every subsequence (akn) of (an) is also bounded.
  • If a sequence (an) is upper (or lower) bounded then every subsequence (akn) of (an) is also upper (or lower) bounded.
  • If at least one of the subsequences of (an) is not bounded then (an) is also not bounded.
  • If at least one of the subsequence of (an) is not upper (or lower) bounded then (an) is also not upper (or lower) bounded.
  • If a sequence (an) is strictly monotonic (increasing or decreasing) then every subsequence (akn) is also strictly monotonic (increasing or decreasing) .

Sequence Convergence:

A sequence (an) is called a null sequence when lim n -> +∞ (an) = 0.

For example the sequence a(n) = 1/n is strictly increasing, but also a null sequence.

So, the convergence has to do with the limit to infinity.

     If the limit is equal to a real number l then we say that the sequence converges (is convergent) and this number l is called the limit of the sequence. So, lim n -> +∞ (an) = l . When the limit doesn't exist or equals +-∞ then the sequence deverges (is devergent).


Examples:

1. a(n) = 2 + 1/n

lim n -> +∞ a(n) = lim n -> +∞ [2 + 1/n] = 2.

So, the sequence converges to 2.


2. b(n) = (4n^2 -3n + 4) / (n^2 +1)

lim n -> +∞ b(n) = lim n -> +∞ [ (4n^2 -3n + 4) / (n^2 +1)] = 4/1 = 4.

Cause this is a polynomial limit with same degree on numerator and denominator.

So, the sequence converges to 4.


Properties:

  • The limit l of a sequence (an) is unique if it exists (sequence is convergent).
  • If a sequence (an) is convergent then it also is bounded. The opposite is not true!
  • If a sequence (an) is not bounded then it also doesn't converge.
  • If a sequence (an) converges to l then every subsequence (akn) of (an) also converges to l.
  • If 2 or more subsequences of (an) converge to a different number l then the sequence (an) doesn't converge.
  • If a sequence (an) gets cut down to many subsequences that converge to the same limit l, then l is also the limit of (an).

If (an) and (bn) are 2 convergent sequences then:

  1. lim n -> +∞ [a(n) +- b(n)] = a + b, where a and b are the limits of (an) and (bn).
  2. lim n -> +∞ [a(n) */ b(n)] = a */ b, where a and b are the limits of (an) and (bn).
  3. lim n -> +∞ [1/a(n)] = 1/a, with a!=0 and a(n)!=0 for every natural n. 
  4. lim n -> +∞ [c*a(n)] = c * a, with c being a real number and a the limit of (an)

The same properties can be applied to more than 2 sequences that converge.

Some more things that we can proof:


If (an) is a null sequence and (bn) is bounded then:

lim n -> +∞ [a(n) * b(n)] = 0 


If (an) is monotonic and bounded then:

  • If (an) is strictly increasing then it also is upper bounded and converges to the supremum bound
  • If (an) is strictly decreasing then it also is lower bounded and converges to the infimum bound


If(an) is a convergent sequence with limit a then:

  • the absolute sequence |(an)| converges to |a|. The opposite is not true!
  • The limit of the k-root of |(an)| equals the k-root of |a|, for a natural number k.


Squeeze Theorem or Sandwich Theorem/Rule for sequences:

Suppose the sequences (an), (bn) and (cn) with bn <= an <= cn for every natural number n.

    If (bn) and (cn) converge to the same limit l in R and so lim n -> +∞ b(n) = l = lim n -> +∞ c(n) then the limit of the sequence (an) is also equal to l. So, lim n -> +∞ a(n) = l.


Convergence Criterion:

    If (an) and (bn) are sequences with |a(n)| <= |b(n)| for every natural n, then if (bn) a null sequence then (an) is also a null sequence and so converges.



And this is actually it for today and I hope you enjoyed it!

Next time we will talk about some special and devergent sequences.

Until next time...Bye!

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