Convergence Tests

Question 1

Limit Comparison Test

Answer 1

L = lim|ak/bk|
k->∞
and L > 0 and finite, then both series converge or diverge

Question 2

Comparison Test

Answer 2

Comparison test
Guess whether Σµk converges
Find a series that proves this (either a smaller one that diverges or a bigger one that converges)

Question 3

Ratio Test

Answer 3

L = lim|uk+1/uk|
k->∞

If L < 1, the series converges
If L > 1 or L=∞, the series diverges
If l=1, the series may converge or diverge

Question 4

Root Test

Answer 4

L = lim kõk
k->∞

If L < 1, the series converges
If L > 1 or L=∞, the series diverges
if L=1, the series may converge or diverge; another test must be tried

Question 5

Integral Test

Answer 5

Find a corresponding function.
Both Σµk and ∫1->∞ f(x)dx both converge or diverge

Question 6

AST

Answer 6

1) an > an+1 2) lim ak = 0
k->∞

Question 7

Geometric Series

Answer 7

Σa(r)^n-1 = a/1-r for |r| < 1.
converges if -1 < r < 1

Question 8

Divergence

Answer 8

lim µk = 0, then the series Σµk may converge or diverge.
k-> ∞

lim µk ≠ 0, then Σµk diverges.
k-> ∞

Question 9

P-series

Answer 9

P-series is of the form:
Σµk = 1/k^P
or any constant instead of 1.
If P>1, the series converges. if 0<P≤1, the series diverges.