series and Sequences

a series converges if…

the limit exists

a sequence is monotonic if …

increasing or decreasing

integral test

if f is continuous, positive, and decreasing,

evaluate the indefinite integral of f(x) where f(x)= a_n, replacing n with x.

Both the series will converge or diverge depending on result.

Geometric series

ar^n

converges when 0<r<1

P series

1/n^p

Converges when p>1

Harmonic series

1/n, type of p-series

Diverges

A series converges if lim as n approaches infinity of sequence

Equals to 0

Direct comparison test

Pick a series you know the behavior of

Determine if it’s lower or upper by setting up an inequality

If upper series converges, so does lower series

if lower series converges, so does upper series

Limit comparison test

(you get to pick bn, preferably a series whose behavior you know, usually a p-series)

If an and bn are positive

L= Limit of an/bn

If L is positive, both diverge/both converge

Alternating series

+,-,+,-,…

Converges if lim=0

AND

an is decreasing

Determine this by assuming an+1<an

Simply replace all n’s in a series by n+1

Set up inequality and simplify till you can say for sure if an+1<an aka if an is decreasing

Absolute converges

If |an| converges, so must an

Split series up | | | | and analyze

Ratio test

an=/0

If lim |an+1/an|

<1, converges absolutely

>1, diverges

=1, inconclusive

Root test

an=/0

Lim nth root of |an|

<1 converges absolutely

>1 diverges

=1, inconclusive