Sequence

What defines a sequence?

A sequence is a function from positive integers to real numbers.

How is a sequence denoted?

Standard notation uses braces, like {an}.

What happens if a sequence diverges to infinity?

Terms exceed any fixed number as n increases.

What is the limit of a convergent sequence?

The limit is unique for convergent sequences.

What does it mean for a sequence to converge?

It approaches a specific number as n increases.

How do you show a sequence diverges?

Demonstrate terms do not settle within any interval.

What is the implication of the Sandwich Theorem?

If an ≤ b ≤ cn and limits of an and cn are equal, then limit of b is also equal.

What does lim (an + bn) equal?

It equals the sum of their limits, a + b.

What is the result of lim (kan)?

It equals k times the limit of an.

What does it mean if a sequence diverges?

No limit exists that the sequence approaches.

What is the condition for a sequence to converge?

For every ε > 0, there exists an N such that |an - L| < ε for n > N.

What happens to the sequence 1, -1, 1, -1, ...?

It diverges because it does not settle at any limit.

What is the limit of √(n+1)/n as n approaches infinity?

It converges to 1.

What does lim (1 + 1/n)^n equal?

It converges to e as n approaches infinity.

What is the limit of n^p as n approaches infinity for p > 0?

It diverges to infinity.

What is the limit of n!/(n^n) as n approaches infinity?

It converges to 0.