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 as signify?
The terms get arbitrarily close to for large .
What does a finite limit imply about a sequence?
That the sequence converges.
When does the sequence converge?
Precisely when .
What is the limit of if ?
The limit is zero.
What is the limit of when ?
The limit is one.
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