Calc 2 Tests for Convergence and Divergence

Geometric Series Test

The geometric series is given by
sum a*r^n = a + a r + a r2 + a r3 + ...

--> If |r| < 1 then the following geometric series converges to a / (1 - r).
--> If |r| >= 1 then the above geometric series diverges.

n-th Term Test

--> If the limit of a[n] is not zero, or does not exist, then the sum diverges.

Integral Test

--> If for all n >= 1, f(n) = an, and f is positive, continuous, and decreasing then either both converge or both diverge.

P-Series Test

--> The p-series is given by
sum 1/n^p
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.

Comparison Test

If 0 <= an <= bn for all n greater than some positive integer N, then:
--> If sum bn converges, then sum an converges.
If sum an diverges, then sum bn diverges.

Limit Comparison Test

If lim (an / bn) = L,
where an and bn > 0 and L is finite and positive,
then the series sum an and sum bn either both converge or both diverge.

Alternating Series Test

Suppose that we have a series and either or where for all n. Then if lim(bn) = 0 and bn is a decreasing sequence, the series is convergent.

Ratio Test

If for all n, (n not equal to 0), then the following rules apply:
Let L = lim | an+1 / an |
If L < 1, then the series sum an converges.
If L > 1, then the series sum an diverges.
If L = 1, then the test in inconclusive.

Absolutely Convergent

A series sum(an) is called absolutely convergent if sum(|an|) is convergent.
***difference between an and |an| convergence is |an| you choose bn (ex: |(-1)^n / n| = 1/n --> diverges by integral test)

Conditionally Convergent

If sum(an) is convergent and sum(|an|) is divergent we call the series conditionally convergent.