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:

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