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Sequences, Relative Growth Rates
Chapter 8, Kennedy, Waits, Demaney
8
Mathematics
12th Grade
02/16/2021

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Term
What is a sequence?
Definition
A sequence is a list of numbers. It is also a function whose domain is a subset of non-negative integers.
Term
What is the explicit formula for the general term of an arithmetic sequence?
Definition
[image]
Term
What is the explicit formula for the general term of a geometric sequence?
Definition
[image]
Term
What is are the two ways an integral could be improper?
Definition
An integral is improper if one or both of the limits of integration are infinite or if a vertical asymptote exists between the limits of integration.
Term
Describe the steps for evaluating the integral:
[image]
Definition
1) Rewrite as the limit:
[image]

2) Evaluate the definite integral.

3) Evaluate the limit.

4) If the limit exist, the integral converges to the limit. If the limit DNE for any reason, the integral diverges.
Term
Describe the steps for evaluating the integral:
[image]
Definition
1) Split into two integrals:
[image], where c is ANY constant (pick one).

2) Evaluate ONE of the two integrals.

3) If either of the chosen integrals diverges, then the original diverges.

If both integrals converge, then the original integral is the sum of the two convergent integrals.
Term
Describe the steps for evaluating the integral:[image]
Definition
1) Split the integral into two:
[image], where f(a)=0 (where the integrand has a vertical asymptote).

2) Evaluate the integrals one at a time.

3) If one of the integrals diverges, the original diverges.

If both converge, then the original converges to the sum of the two convergent integral
Term
Explain how to determine whether f(x) or g(x) grows at a faster rate.
Definition
1) If [image], then f(x) grows faster.

2) If [image], f(x) grows slower (g(x) grows faster).

3) If [image], f and g grow at the same rate.
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