Integral Test 3

Conditions for Improper Integral

At least one of the limits of integration in infinity or there is on a vertical asymptote

When does a sequence converge

If it is monotonic and bounded

When does a sequence diverge

if it is unbounded

Geometric series form

r^n

When does a Geometric series converge and diverge

a geometric series converges to 1/(1-r) when |r|<1, diverges otherwise

P-Series form

1/k^p

When does a p-series converge or diverge

converges when p>1, diverges when 0<p<=1

How do you show results of a p-series

integral test

Harmonic Series form

1/n

When does a harmonic series converge or diverge

it always diverges

how do you find the sum of a telescoping series

use partial fractions

The series ∑an diverges if the limit is NOT equal to

0

If lim n→∞ an=0, then what do we know about the series ∑an?

Test is inconclusive, try another test

Conditions for Integral Test

positive, continuous, decreasing

If you want to show a series converges, compare it to a _________ series that also converges.

larger

if you want to show a series diverges, compare it to a _______ series that also diverges.

smaller

If the direct comparison test does not have the correct inequality, you use the _____

limit comparison test

conditions for limit comparison test

find a comparison series and state whether the comparison converges or diverges

math for limit comparison test

L=lim n→∞ (an)/bn

ln(1)

0