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 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.
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:
If (an) and (bn) are 2 convergent sequences then:
The same properties can be applied to more than 2 sequences that converge.
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 a convergent sequence with limit a then:
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!
