Pure Mathematics D2: Sequences

50 flashcards · Mathematics · by lakesare

What's the definition of a sequence.
Sequence – an unending list of real numbers.
What's a shorter way to denote this sequence?
How to expand this notation?
How can you represent a sequence as a function?
One helpful way to think of a sequence is as a function f with domain N and codomain R that maps each number n in the domain to the nth term of the sequence.
Visually imagine the sequence diagram.
Visually imagine: increasing sequence VS strictly increasing sequence.
increasing sequence strictly increasing sequence
What's bizarre about constant sequences?
They are both increasing and decreasing.
When exactly is (an) dominated by (bn)? (bn) is a sequence of non-negative terms.
When (an) is always below (bn).
How can we describe this sequence diagram?
It's eventually decreasing. It's also eventually strictly decreasing.
When should you use Strategy D3 VS D4?
Use either strategy, depending on what's easier to simplify.
[Null sequences] What's a null sequence in terms of a limit?
A sequence that has zero as its limit is sometimes called a null sequence.
[Null sequences] Why is an a null sequence, in the ε-N definition?
Because for any ε, we can find N such that all dots fall within the strip (for n > N). Null sequence is when x = 0.
[Null sequences] Who would win the ε-N game if (an) is a null sequence? Each player makes their choices carefully.
Player B, as there's always a number N such that all n > N fall within the strip. Notice that in the 'ε-N game', if (an) is any null sequence and both players make their choices carefully, then player B will always win.
[Convergent sequences] When is the sequence convergent VS divergent?
convergent - when an tends to a real number limit when n → ∞. divergent - when it's not convergent.
[Null sequences] Who would win in this ε-N game? ε-N game
Player A (because Player B wouldn't be able to come up with N). We can see that 2 would not be a suitable value of ε but 1/2 would be a suitable value. In our 'ε-N game', if player A plays ε = 1/2 , then there is no integer N that player B can play to win.
[Null sequences] Visually imagine a null sequence.
The sequence an = 1/n
[Null sequences] What kind of number is ε?
Positive real number.
[Null sequences] Why is it so that "the sequence (an) is null ↔ the sequence (|an|) is null"?
Because the definition considers (|an|) anyways.
[Null sequences] What do we have to find to show that (an) is a null sequence (using the original definition)? The sequence an = 1/n Null sequence is when x = 0.
We want to express N (usually) in terms of ε.
[Null sequences] What's the point of these rules?
Knowing (an) is a null sequence, we can conclude some other sequence is a null sequence.
[Null sequences] What's the general approach for proving these rules?
We notice that if it works for ε, then it works for ε/|λ| and then just rearrange the inequality.
[Null sequences] Visually imagine the squeeze rule.
an is squeezed between bn and the x-axis. and thus an is a null sequence, just like bn.
[Null sequences] Give an example of a dominating null sequence for an.
In this case, we know that −1 ≤ cos(2n) ≤ 1 by the properties of the cosine function. This suggests that (2/n2) might be a suitable dominating sequence.
What's the "limit of a sequence" in lim notation?
[Null sequences] Why do we care about the basic null sequences?
Using basic null sequences, we can deduce many other sequences are null (without applying the definition).
[Convergent sequences] How would you define a sequence convergent with limit L using the null sequence definition? Null sequence definition
(an) is convergent with limit L ≡ (an - L) is a null sequence.
[Convergent sequences] How do you read this?
The limit of an, as n tends to infinity, is L.
[Convergent sequences] What's another way to write this?
[Convergent sequences] In what cases do we say: a tends to b VS a converges to b.
n tends to ∞ an converges to L
[Convergent sequences] Visually imagine the sequence converging to L definition.
A sequence which converges to L. For each positive number ε, the terms an eventually lie inside the horizontal strip from L − ε to L + ε.
[Convergent sequences] Visually imagine the convergent VS divergent sequence.
Convergent sequence Divergent sequence
[Convergent sequences] What's the connection between: combination rules for null sequences, and combination rules for convergent sequences?
rules for null sequences - (an + bn) results in a sequence that converges to 0 + 0. rules for convergent sequences - (an + bn) results in a sequence that converges to l + m. These are same, rules for convergent sequences are just more general.
[Convergent sequences] How is it so that Quotient Rule works even though bn is sometimes 0?
bn = 0 only for finitely many bn.
When is the sequence monotonic?
When it's always either increasing or decreasing.
State the hierarchy of these terms: np n! cn. As n → ∞.
n! (strong) cn np (weak)
How would you remember the basic null sequences?
Just remember the hierarchy of terms: n! (strong) cn np (weak)
Name 2 steps to find the limit of this sequence.
Divide by the dominant term Apply the combination lim rules Just remember the hierarchy of terms: n! (strong), cn, np (weak).
Visually imagine the Squeeze Rule.
What's an interesting catch about the Limit Inequality rule?
Taking limits doesn't preserve strict inequalities!
Divergent VS detergent?
Divergent - e.g. a divergent sequence A sequence is divergent if it is not convergent.Detergent - mixture with cleansing properties.
Visually imagine a bounded sequence. Thus a sequence (an) is bounded if all the terms an lie on the sequence diagram in the horizontal strip from −M to M , for some positive number M.
Visually imagine a sequence that tends to infinity.
Goes infinitely up!
Name 2 criteria to determine that the sequence is divergent. We give a strategy for proving divergence using two criteria, which together cover all cases. We obtain these criteria by establishing certain properties which are necessarily possessed by a convergent sequence; if a sequence does not have one of these properties, then it must be divergent.
(an) has 2 convergent subsequences with different limits (an) has a subsequence → ∞ or –∞ To apply Strategy D8 successfully, you need to be able to recognise convergent subsequences with different limits, or a subsequence (which may be the whole sequence) which tends to infinity or to minus infinity. It is not always easy to do this, and some experimentation may be required.
What do these notations (a2k) VS (a2k – 1) mean?
(a2k) - the "even subsequence" of an. (a2k – 1) - The "odd subsequence" of an.
State this theorem in your own words.
If lim an = l, then lim of a more rarely sampled an = l, too.
What does it mean to say that (an) consists of 2 subsequences?
It means that every term of the sequence belongs to one of these sequences.
[Monotone Convergence Theorem] Visually imagine the Monotone Convergence Theorem.
An increasing sequence that is bounded above. A decreasing sequence that is bounded below. Monotone Convergence Theorem states that any sequence that is: increasing AND bounded above must be convergent.
[Monotone Convergence Theorem] Name 2 uses for the Monotone Convergence Theorem. An increasing sequence that is bounded above.
Prove the sequence is convergent The Monotone Convergence Theorem is most useful when we suspect that a sequence is convergent, but we cannot find the limit directly.Define π It can also be used to give precise definitions of numbers, such as π, about which we have only an informal idea, as you will see in the next subsection.
Visually imagine the 1st Subsequence Rule.
Visually imagine the 2nd Subsequence Rule.