Convergent Sequences and Limit Theorems
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Questions and Answers

What can be concluded if a sequence is both Cauchy and convergent?

  • The sequence has two different limits.
  • The sequence is monotonic.
  • The sequence is necessarily divergent.
  • The sequence is bounded. (correct)

When is a sequence considered monotonic?

  • If it has a limit of zero.
  • If the sum of its terms diverges.
  • If all its terms are distinct.
  • If it consistently increases or decreases. (correct)

Which sequence is divergent?

  • The constant sequence $s_n = 3$.
  • The sequence defined by $s_n = n$. (correct)
  • The sequence defined by $s_n = rac{1}{n}$.
  • The sequence defined by $s_n = (-1)^n$. (correct)

If a sequence is convergent, which of the following must be true?

<p>It is a Cauchy sequence. (A), It has a unique limit. (C)</p> Signup and view all the answers

What is a key property of the subsequences of a convergent sequence?

<p>They converge to the same limit as the original sequence. (B)</p> Signup and view all the answers

Which statement is true regarding the sequence (sn) = ((−1)n)?

<p>The subsequences converge to different limits. (A)</p> Signup and view all the answers

According to the theorems presented, what can be inferred about a convergent sequence?

<p>Every convergent sequence must have a limit that exists. (C)</p> Signup and view all the answers

What does it mean for a sequence to be unbounded?

<p>The sequence does not maintain a maximum or minimum value. (A)</p> Signup and view all the answers

Which of the following is a direct result of using the triangle inequality in proving limits?

<p>It helps bound the difference between the sequence product and the product of the limits. (A)</p> Signup and view all the answers

If a sequence (sn) is both bounded and contains subsequences converging to different limits, what can be concluded?

<p>The sequence is divergent. (C)</p> Signup and view all the answers

Flashcards

Divergent Sequence

A sequence that does not have a finite limit. It can either oscillate infinitely or grow infinitely large (or small).

Convergent Sequence

A sequence that approaches a specific finite value as the number of terms increases infinitely.

Bounded Sequence

A sequence is bounded if all its terms lie within a specific range. There's a maximum and minimum value that all terms fall between.

Unbounded Sequence

A sequence is unbounded if its terms can get arbitrarily large (or small) without any limit.

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Boundedness and Divergence

If a sequence converges, it must be bounded. This means that if a sequence is unbounded, it must diverge.

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Convergence of a sequence (sn)

A sequence (sn) converges to a real number s if for any positive number ε, there exists a natural number N such that for all n greater than N, the absolute difference between sn and s is less than ε. This means that the terms of the sequence get arbitrarily close to s as n approaches infinity.

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Constant Sequence

The sequence (sn) = (s) is a constant sequence where every term is equal to s. The limit of this sequence is s, meaning it converges to s.

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Limit of 1/n as n approaches infinity

The sequence (sn) = (1/n) converges to 0. This means that as n gets larger, the terms get closer and closer to 0.

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Subsequence Convergence

A subsequence of a sequence (sn) is a sequence formed by selecting some terms from the original sequence, keeping their relative order. If the original sequence converges to a limit, its subsequences also converge to the same limit.

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Study Notes

Convergent Sequences

  • A sequence of real numbers (sn) converges to a real number s if for every ε > 0, there exists an N ∈ â„• such that for all n > N, |sn - s| < ε.
  • This is written as sn → s or s = limn→∞ sn.
  • If a sequence does not converge, it is divergent.
  • A sequence cannot have two different limits.
  • Convergent sequences are bounded.
  • If sn = s for all n, then lim sn = s.
  • Example 1: sn = s (constant sequence) → lim sn = s
  • Example 2: sn = 1/n → lim sn = 0
  • Example 3: sn = (-1)n (divergent)
  • Example 4: sn = n (divergent)
  • Convergent sequences have a unique limit.

Limit Theorems

  • If (sn) converges to s and (tn) converges to t, then (sn + tn) converges to s + t.
  • If (sn) converges to s and (tn) converges to t, then (sn * tn) converges to s * t.
  • If (sn) converges to s, then (k * sn) converges to k * s, where k is a constant.
  • If (sn) converges to s, then (snm) converges to sm.
  • If (sn) converges to s and s ≠ 0, then 1/sn converges to 1/s (provided sn ≠ 0 for all n).
  • If (sn) converges to s and (tn) converges to t, then (sn - tn) converges to s - t.

Bounded Sequences

  • Every convergent sequence is bounded.
  • An unbounded sequence is divergent.

Subsequences

  • If (sn converges to s, then every subsequence of (sn) also converges to s.
  • If a sequence has subsequences that converge to different limits, then the sequence is divergent.

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Related Documents

Convergent Sequences PDF

Description

Explore the concepts of convergent and divergent sequences in this quiz. Understand how limits are defined and the theorems that govern their behavior. Test your knowledge with examples to solidify your understanding of sequences and their limits.

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