Sequences and Series Tests

When to use the Root Test

f(x)^g(x)

When to use the Ratio Test

factorials

When to use the Alternating Series Test

entire term is alternating

When to use the p-Test

1/k^p

When to use the Geometric Series Test

ar^k

When to use the Divergence Test

quick check

When to use the Comparison Tests

break up into leading order

What are the requirements for the Integral Test

continuous, positive, and decreasing over interval

What are the requirements for the Alternating Series Test

positive and decreasing

What is the conclusion for an Alternating Series Test

lim = 0: converges, else: inconclusive

What is the conclusion of the Root and Ratio Tests

r<1: converges, r>1: diverges, r=1: inconclusive

What is the conclusion of the p-Test

p>1: converges, else: diverges

What is the conclusion of the Divergence Test

lim=inconclusive, else: diverges

When to use Telescoping

combining 2 different k terms

What is the conclusion of Telescoping

0 + final gives convergence term

What is the conclusion of the Integral Test

result of integral=result of series

What is the conclusion of the Direct Comparison Test

greater term converges: both converge, lesser term diverges: both diverge, else: inconclusive

What is the conclusion of the Limit Comparison Test

lim=0: new converges implies old converges/old diverges implies new diverges, lim=infinity: new diverges implies old diverges/old converges implies new converges, positive number: converge or diverge together