Series Flashcards
When does the geometric series ar^(n-1) converge?
when |r| < 1
What is the sum of the geometric series
a/1-r
What is the comparrison test?
If 0 <= an <= bn:
if sum bn converges, then sum an converges
if sum an diverges, sum bn diverges
When is sum 1/n^a convergent?
When a > 1
What is the ratio test?
|an+1/an| –> L
L < 1, sum an converges
L > 1 sum an diverges
When is the comparison test useful?
When an is a polnomial/trig function/can easily be bounded from above
When is the ratio test used?
When an contains exp or factorial terms
What does absolutely convergent mean?
if sum |xn| converges, then sum xn converges
What is the root test?
|an|^1/n –> L
L < 1 sum an converges
L > 1 sum an diverges
When is the root test used?
When an is the form bn^1/n
What does n^(1/n) converge to?
1
When does r^(1/n) converge and what to?
When r > 0, converges to 1
What is the radius of convergence for the power series sum an.x^n?
|an+1/an| –> beta
or |an|^1/2 –> beta
R = 1/beta
How do we interpret the radius of convergence?
|x| < R , series converges
|x| > R, series diverges
