Sequence Tests

Question 1

How does P-Series work?

Answer 1

1/(n^p)
P>1 Converges
P<=1 Diverges

Question 2

How does the geometric series work?

Answer 2

If the abs of r>=1 diverges
If the abs of r<1, it converges using the equation (first term)/(1-r)

Question 3

How does P for the R tests work?

Answer 3

Ratio and Root
P=1 inconclusive
P>1 divergent
0<=P<1 convergent
P=infinity divergent

Question 4

How do the R tests, divergence, and integral* tests work?

Answer 4

We take the limit of An and whatever manipulation needs to be done
*integral we take the limit and integral

Question 5

Can the divergence test prove convergence?

Answer 5

No; if An=0 it’s inconclusive and if it doesn’t equal zero it diverges

Question 6

For the integral test if the integral converges does the summation converge?

Answer 6

Yes and vice versa

Question 7

How does the comparison test work?

Answer 7

You take the An term over the Bn term and solve
You can use L’H after???
If Bn converges, An will converge
If An>=Bn, An diverges as welll as Bn

Question 8

How does the absolute comparison test work?

Answer 8

If the abs if An converges, An will converge
P<infinity converges???

Question 9

Comparison LIMIT…

Answer 9

Take the limit of the abs of An/Bn which =L
If L = 0 Bn converges and An converges
If L = infinity Bn diverges and An diverges
If L is finite, then if An converges Bn will converge; and if An diverges Bn will diverge

Question 10

What does (n+1)! =

Answer 10

(n+1)(n!)

Question 11

(n+1)^(n+1) =

Answer 11

(n+1)^n * (n+1)^1

Question 12

[(n)/(n+1)]^-n = as well

Answer 12

[(n+1)/(n)]^n

Question 13

Divergence test…

Answer 13

Basically l’hopitals it till you get a answer. If An=0 converges
If An doesn’t equal zero, series diverge

Question 14

Divergence test…

Answer 14

Basically l’hopitals it till you get a answer. If An=0 converges
If An doesn’t equal zero, series diverge

Question 15

What does An equal for the divergence test to prove convergence

Answer 15

An=0