The sequence converges precisely when , with limit for and limit for .
The Monotonic Sequence Theorem states that every bounded monotonic sequence converges, specifically an increasing sequence bounded above or a decreasing sequence bounded below.
1. What distinguishes a sequence from a series?
2. What does the statement $a_n\to L$ as $n\to\infty$ mean?
3. For which values of $r$ does the sequence $\{r^n\}$ converge?
What is an infinite sequence in mathematics?
An ordered list of numbers indexed by specified positive integers.
What does the notation \(a_n \to L\) as \(n \to \infty\) signify?
The terms get arbitrarily close to \(L\) for large \(n\).
What does a finite limit imply about a sequence?
That the sequence converges.
When does the sequence \(\{r^n\}\) converge?
Precisely when \(-1 < r \leq 1\).
What is the limit of \(\{r^n\}\) if \(-1 < r < 1\)?
The limit is zero.
What is the limit of \(\{r^n\}\) when \(r = 1\)?
The limit is one.
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