Chapter 9 Flashcards
(15 cards)
Recursive Arithmetic Sequence
Arithmetic sequence where the term is determined by the previous term:
An = An-1 + d
Explicit Arithmetic Sequence
Arithmetic sequence where the term is determined by the number of the term
Recursive Geometric Sequence
Geometric sequence where the term is determined by the previous term:
An = An-1 * r
Explicit Geometric Sequence
Geometric sequence where the term is determined by the number of the term
Limit of a Sequence(converging vs diverging)
If the sequence has a limit L as n approaches ∞, then it converges to L. If the sequence has no limit then it diverges.
L’Hopital’s Rule(First Form)
lim(x–>a) = f(x) / g(x) = f’(x) / g’(x)
L’Hopital’s Rule(Stronger Form)
lim(x–>a) f(x) / g(x) = lim(x–>a) f’(x) / g’(x)
When working with indeterminate forms ∞/∞, ∞*0, ∞-∞
Use L’Hopital’s Rule
When working with indeterminate forms 1^∞, 0^0, ∞^0
Take limit of ln of the function(Use L’Hopital’s Rule) and when determined raise e to the limit. This undoes taking the ln.
f(x) grows faster than g(x) as x–> ∞ if:
lim(x–>a) f(x) / g(x) = ∞ or,
lim(x–>a) g(x) / f(x) = 0
f(x) and g(x) grow at the same rate as x–> ∞ if:
lim(x–>a) f(x) / g(x) = L ≠ 0
Transitivity of Growing Rates
If f grows at the same rate as g as x–> ∞ and g grows at the same rate as h as x–> ∞ then f grows at the same rate as h as x–> ∞
If f(x) is continuous on [a, ∞), then ∫(a, ∞) f(x)dx =
lim(b–>∞) ∫(a, b) f(x)dx
If the limit is finite the improper integral converges and the limit is the value of the improper integral. If the limit DNE then the improper integral diverges.
Convergence Comparison Test
If 0 ≤ f(x) ≤ g(x)
∫(a, ∞) f(x)dx converges if ∫(a, ∞) g(x)dx converges
Divergence Comparison Test
If 0 ≤ f(x) ≤ g(x)
∫(a, ∞) f(x)dx diverges if ∫(a, ∞) g(x)dx diverges
