AP Calculus BC - 10.1 - Sequences

How do you test if a sequence converges?

By taking the limit of the function that represents the general term of each term in the sequence. If the limit exists, then the sequence converges.

How do you test if a sequence diverges?

Test for convergence as normal. If the limit doesn’t exist, or goes to either infinity or negative infinity, then the sequence diverges.

What is a geometric sequence?

It's a sequence, represented by a_n = cr^n, where c and r are nonzero

Do the basic limit laws apply to sequences?

Yes

Explain how the squeeze theorem for sequences works, and when you should apply the squeeze theorem.

First of all, you need to apply the squeeze theorem for sequence if taking the limit normally doesn’t work out. You then identify the problem child, find its upper and lower bounds, then use algebra to get from the problem child to the function representing the general term. Obviously, whatever operations you perform on the function should also be done to the bounds as well. Let a_n define the general term, and let b_n and c_n represent the lower and upper bounds respectively. If lim(n → infinity) (b_n) = lim(n→infinity)(c_n) = L, then lim(n→infinity)(a_n) = L

If we have a function inside of a function, and we’re trying to take the limit of it, how can we easily approach it?

Define f(x) as the outer function. If f(x) is continuous, then we assume that the inner function a_n has a limit lim(n→infinity)(a_n) that exists and converges to L. Then we know that the setup of the whole function is f(a_n), and taking the limit of that is simply just f(lim(n→infinity)(a_n)) which is f(L)

When do we say a sequence is bounded above? When is it bounded below?

A sequence is bounded above if there is a value M such that a_n <= M for all n.

A sequence is bounded below if there is a value M such that M <= a_n for all n.

When do we say a sequence is bounded?

When it’s bounded both above and below

When is a sequence monotonic and how does this information help us?

A sequence is monotonic if it’s increasing (each value is greater than the previous value), or decreasing (each value is smaller than the previous value)

Sequences that are both monotonic and bounded converge