Test Series Conditions of

Convergence

Conditions of

Divergence Comment

nth-Term 1

n

n

a∞

=

∑ lim 0nn

a→ ∞

≠ This test cannot be

used to show

convergence.

Geometric ( )0

n

n

a r∞

=

⋅∑ 1r < 1r ≥ Sum: 1

aS

r=

−

Telescoping ( )1

1

n n

n

a a∞

+

=

−∑ lim nn

a L→ ∞

= Sum: 1S a L= −

p-Series 1

1p

n n

∞

=

∑ 1p > 1p ≤

Alternating Series ( )1

1

1n

n

n

a∞

−

=

−∑ 10

n na a

+< < and

lim 0nn

a→ ∞

=

nS as estimate of sum

remainder 1n nR a

+<

Integral

(f is continuous,

positive, decreasing) 1

,

( ) 0

n

n

n

a

a f n

∞

=

= ≥

∑

1( )f x dx

∞

∫ converges 1( )f x dx

∞

∫

diverges

1( )

nn

S f x dx∞

++ ∫ and

( )n

nS f x dx

∞

+ ∫ are

bounds for estimation

of sum by n

S .

Direct Comparison

( ), 0n na b > 1

n

n

a∞

=

∑

0n n

a b< ≤ and

1

n

n

b∞

=

∑ converges

0n n

b a< ≤ and

1

n

n

b∞

=

∑ diverges

Useful for series

similar to p-series and

geometric.

Limit Comparison

( ), 0n n

a b > 1

n

n

a∞

=

∑

a) lim ; if 0n

nn

aL L

b→ ∞= < < ∞ they both

behave in the same manner.

b) 0 if n

L b= ⇒ converges then n

a

converges.

c) if n

L b= ∞⇒ diverges then n

a diverges.

Useful if not able to

show 0n n

a b≤ ≤ for

direct comparison

Ratio 1

n

n

a∞

=

∑ 1lim 1n

nn

a

a

+

→ ∞< 1lim 1n

nn

a

a

+

→ ∞>

The test is

inconclusive if

1lim 1n

nn

a

a

+

→ ∞=

Root

(Not on IB) 1

n

n

a∞

=

∑ lim 1nn

na

→ ∞< lim 1n

nn

a→ ∞

>

The test is

inconclusive if

lim 1nn

na

→ ∞=