Test 1 9.1-9.4 Flashcards
A ______ is a function whose domain is the set of positive integers.
sequence
If f is the function and n is a positive integer, then
a_n = f(n)
A sequence is ________ if either (a) if each term is greater than or equal to the preceding term, or (b) each term is less than or equal to the preceding term
monotonic
A sequence {a_n} is ______ _______ if there is areal number N such that N ≤ a_n for all n.
bounded below
The number M is called an _____ _____ for a sequence {a_n} if a_n ≤ M for all n.
upper bound
The _____ of a sequence {a_n} is : if for each ℇ>0, there exists M>0 such that |a_n-L| < ℇ whenever n>M.
Limit
In this case, the sequence _______ to L, and we write a_n → L
converges
Assume that {a_n} converges to L and {b_n} converges to K.
If a sequence is bounded and monotonic then it (converges/diverges)
converges
Assume that {a_n} converges to L and {b_n} converges to K.
{a_nb_n} converges to [(LK)/L+K]
L*K
Assume that {a_n} converges to L and {b_n} converges to K.
{a_n/b_n} converges to (L/K / K/L), if b_n ≠ 0 and L≠0 and K≠0
L/K
If {a_n} and {b_n} both converge to L, and if there exists and N such that for all n>N, a_n ≤ c_n ≤ b_n then {c_n} converges to L.
Squeeze Theorem
The number N is called a _____ _____ for a sequence {a_n} if N≤a_n for all n.
lower bound
A sequence {a_n} is _____ _____ if there is a real number M such that a_n ≤ M for all n.
bounded above
A sequence is said to _______ if it has a limit.
converge
A sequence is said to _______ if it does not have a limit.
diverge
Assume that {a_n} converges to L and {b_n} converges to K.
If lim f(x) = M, as x→∞, and if c_n (= / ≠) f(n), then c_n → M.
=
Assume that {a_n} converges to L and {b_n} converges to K.
{cb_n} converges to ( cK / c ), for any real number c.
c*K
Assume that {a_n} converges to L and {b_n} converges to K.
{a_n +b_n} converges to (L+K / 2L).
L+K
If | c_n | → (0 / M), then c_n converges.
0 (Absolute Value Theorem )
For a sequence {a_n}, the ______ _____ has the form S_n= a1 +a2+a3+…+a_n
partial sum
For any sequence {a_n}, the sequence {S_n} of its partial sums if called the ____ _____
series ∑a_n
A series ________ if and only if its ________ _______ ______ converges; otherwise, the series _______.
Converges
sequence of parital sums
diverges
The limit of a series is called its ____
sum
A _____ ________ has the form (b1-b2)+(b2-b3)+(b3-b4)+….
telescoping series
A _________ _________ has the form a∑r^n = a+ ar+ar^2 +a^3 + a^4….., with the index n going from 0 to ∞. The ______ is _ and the ____ ____ is _.
geometric series ratio r first time a
If the sequence of ( terms / partial sums) of {a_n} converges, then ∑a_n converges
partial sums
If ∑a_n converges, then {a_n} converges to ( xero / a positive constant )
xero
A geometric series with first a and ratio r will converge to a/(1-r), if |r| is ( greater / less ) than 1.
greater
If {b_n} converges to a finite constant, B, then the telescoping series ∑(b_n -b_n+1) converges to ( B /b_1 -B )
B
