Convergent Sequences and Limit Theorems

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

When is a sequence considered monotonic?

Which sequence is divergent?

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

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

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

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

What does it mean for a sequence to be unbounded?

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

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

Convergent Sequences

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

Limit Theorems

If (s_n) converges to s and (t_n) converges to t, then (s_n + t_n) converges to s + t.
If (s_n) converges to s and (t_n) converges to t, then (s_n * t_n) converges to s * t.
If (s_n) converges to s, then (k * s_n) converges to k * s, where k is a constant.
If (s_n) converges to s, then (s_n^m) converges to s^m.
If (s_n) converges to s and s ≠ 0, then 1/s_n converges to 1/s (provided s_n ≠ 0 for all n).
If (s_n) converges to s and (t_n) converges to t, then (s_n - t_n) converges to s - t.

Bounded Sequences

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

Subsequences

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

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.