Tests for Convergence (of Divergence) of a series

1

nn

a

∞=

∑

Name of Test Brief on how it works When to use it

Test for Divergence If lim0

nn

a

→ ∞

≠

, then

1

nn

a

∞=

∑

diverges.

Can be used anytime — it’s a quick check to see if the series diverges Integral Test Find function from nth general term()

n

f n a

=

, if

1.2.3.

f positive f continuous f decreases

Then

()convergesconverges()divergesdiverges

nn

f x dx a f x dx a

⇒⇒

∑∫∑∫

When you have a positive, decreasing series and when ()

f x

is easy to integrate. Often used when logarithms are involved.

Comparison Test

nnn=1n=1nnn=1

(which you know converges or diverges.)

If the given series is , and the chosen series is If 0 and b converges, then aalso converges.If 0 and b diverges, then aalso

nnn nn n

aba ba b

∞ ∞∞

≤ ≤≥ ≥

∑ ∑∑

n=1

diverges.

∞

∑

When you have a positive series that can be compared to another positive series known to converge or diverge. Often choose a p-series to use for comparison. Limit Comparison Test

nnnn=1n=1nnn=1

(which you know converges or diverges.)

If the given series is , and the chosen series is If lim,then b converges then aalso converges b diverges then aalso diver

=#

nnnn

abab

nonzero

→ ∞∞ ∞∞

∑ ∑∑

n=1

ges

∞

∑

When you have a positive series that can be compared to another positive series known to converge or diverge. Preferred to Comparison test, but will be inconclusive if the limit is 0. Alternating Series Test

1n

Determine what is for the given series.If 1.for all n;AND2. If limthen the series converges.

=0

nn n

n

bb b

b

+→ ∞

≤

When you have an ALTERNATING series that is decreasing. Absolute Convergence Test

11

If converges, then also converges.

n nn n

a a

∞ ∞= =

∑ ∑

Abs. Convergence implies convergent

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