Covergence Tests

Question 0

Geometric series test

Answer 0

Of the form ar^n(or n-1)
If -1<1 then the series converges. (Correlation implies convergence)
Otherwise, series is divergent

Question 1

P-series test

Answer 1

Of the form 1/(n^p)
If p>1, the series is convergent
If p<= 1, the series is divergent

Question 2

Direct Comparison Test

Answer 2

If an < bn for all n, and bn converges, then an converges

If an > bn for all n, and bn diverges, then an diverges

Question 3

Limit Comparison Test

Answer 3

Limit |an/bn| = L

If L is finite and > 0, then an and bn either both converge or both diverge.

Question 4

Alternating Series Test

Answer 4

Of the form sigma (-1)^n bn
Where
Lim Bn =0
B(n+1)<=Bn (decreasing)
The series is convergent

Question 5

AST estimation theorem

Answer 5

|Rn| = |s-sn| <= b(n+1)

Question 6

Root test

Answer 6

Lim nth root of |An| = L
If L < 1, absolutely convergent
If L > 1 of infinity, divergent
If L=1, the test is inconclusive

Question 7

Ratio Test

Answer 7

Lim |A(n+1)/An|=L
If L 1 or infinity, divergent
If L=1, inconclusive

Question 8

Integral test

Answer 8

If the corresponding function is integrable, then do this one.
If integral =finite number, then the series converges.
If the integral goes to infinity, the. The series is divergent.