Tests for Series Convergence
Jefferson Davis Learning Center, Sandra Peterson
∞
Suppose you have a series that looks like this:
∑ an
n =1
Begin by finding: lim a n ≠ 0
n →∞
Nth Term Test
Limit does not equal zero or does not exist?
No
Yes
The series either converges or diverges.
More tests are needed.
Do you have all positive terms or alternating terms?
The series diverges.
All positive terms
Alternating terms
Alternating Series Test
Try one of the tests listed below.
Decreasing
Increasing or ???
The series converges.
The series diverges.
If your series has all POSITIVE terms, then you should try one of the following tests:
I.
The Geometric Series
∞
∑ ar n
where a and r are constants.
n =1
A.
If r ≥ 1 , then the series diverges.
B.
If r < 1 , then the series converges. The sum is S =
a
.
1− r
II.
The P-Series
∞
1
∑ k p where p is a constant.
k =1
A.
If 0 < p ≤ 1 , then the series diverges.
B.
If p > 1 , then the series converges.
III.
The Comparison Test (for positive terms only)
Let ∑ a k and ∑ bk be two series such that a n ≤ bn for all n .
A.
B.
If ∑ bk ("bigger" series) converges, then ∑ a k ("smaller" series)
converges.
If ∑ a k ("smaller" series) diverges, then ∑ bk ("bigger" series)
diverges.
IV.
The Limit Comparison Test (for positive terms only)
a
Let ∑ a k and ∑ bk be two series. And let ρ = lim k .
k → +∞ b
k
If ρ is finite and ρ > 0 , then the series both:
A.
converge (smaller than a convergent series converges)
B.
OR diverge (bigger than a divergent series diverges)
V.
The Ratio Test (for positive terms only)
Let
A.
B.
C.
VI.
be a series and suppose that ρ = lim
k →+∞
u k +1
.
uk
If ρ < 1 , the series converges.
If ρ > 1 or if ρ = +∞ , then the series diverges.
If ρ = 1 , then the series may converge or diverge. You must use
another test.
The Root Test (for positive terms only)
Let
A.
B.
C.
VII.
∑ uk
∑ uk
be a series and suppose that ρ = lim
k
k →+∞
u k = lim (u k ) k .
i
k → +∞
If ρ < 1 , the series converges.
If ρ > 1 or if ρ = +∞ , then the series diverges.
If ρ = 1 , then the series may converge or diverge. You must use
another test.
The Integral Test (for positive terms only)
Let ∑ u k be a series and suppose that f (x ) is the function that results when
k is replaced with x in the formula for u k . If f (x ) is decreasing and
continuous on the interval [a, + ∞ ), then
both converge or both diverge.
∞
∑ uk
k =1
and
+∞
l
a
a
f (x ) dx
∫ f (x ) dx = llim
→ +∞ ∫