Sequences and Series Flashcards
When does a sequence converge?
Limit of the sequence as n –> infinity exists (not infinity/negative infinity)
What is the squeeze theorem?
When there are two functions, one higher and one lower than another function for all values of x. If both the higher and lower funcitons have the same limit, then the middle function also goes to this limit.
Absolute value theorem
For sequences
the limit of the absolute value of a sequence is equal to the limit of the sequence.
B+M
For sequences
If a sequence in bounded (both sides) and monotonic (not increaseing or not decreasing), then it converges.
Geometric Series Test
if the change multiplyer (r) >= 1, then diverge. r < 1, then it converges.
Geometric Series Formula
first term/(1 - rate of change)
Divergence Test
if the limit as n –> infinity of a(n) doesn’t go to zero, then the series converges.
How to solve
Telescoping Series
Use partial sums
Integral Test
If f(x) is positive, continuous nad decreaisng for x>= a certain value and a(n) = f(n), then the series and integral either both converge or both diverge.
P-Series
if 0< p <= 1, then it diverges. If p > 1, then it converges.
Direct Comparison Test
If a sequence S1(n) converges and 0 < S2(n) <= S1(n) for all values past a certain point, then S2(n) converges.
Limit Comparison Test
If a(n) >0 and b(n) > 0, and lim as n –> infinity (a(n)/b(n)) = L where L is finite and positive, then the two series either both converge or both diverge.
Alternating Series Test
if a(n) > 0, lim as n –> infinity a(n) = 0 and a(n) decreases monotonically, then the series with a(n) and an alternating bit converges conditionally.
Alternating Series Error Test
S(n) will approximate the limit for infinite convergence with error no greater than a(n+1).
Absolute Convergence Test
If sum abs(a(n)) converges, then sum a(n) converges absolutly.
