Math 1B
§ 11.7 Strategy for Testing Series
1.
If you can see at a quick glance that lim an ≠ 0 , use the Test for Divergence.
2.
If the series is of the form
n →∞
1
∑n
€
series is of the form ∑ ar or
∞
n
n= 0
, it is a p-series (convergent if p > 1, divergent if p ≤ 1). If the
∞
∑ ar
n−1
, it is a geometric series (convergent if r < 1, divergent if
n=1
€
€
r ≥ 1). Often some
€ algebraic manipulation is required to get a geometric series into the correct
form.
€
€
€
If the series is similar to a p-series or a geometric series, try the Comparison Test.
3.
€
p
4.
If the series is a rational expression involving only polynomials or polynomials under radicals,
try the Comparison Test and/or Limit Comparison Test and compare with a p-series. (The value
of p should be chosen by keeping only the highest powers of n in the numerator and
denominator.)
Note: The Comparison Test and Limit Comparison Test apply only to series with positive terms.
But if ∑ an has some negative terms, we can apply the Comparison Test to ∑ an and test for
absolute convergence.
5.
€
€
€
If the series contains factorials or constants raised to the nth power, try the Ratio Test.
€
∑ (−1)
n−1
bn or
∑ (−1)
n
6.
If the series is of the form
bn , try the Alternating Series Test.
7.
If an is of the form (bn ) , try the Root Test.
8.
€
€
If an = f (n) for some positive, decreasing function and
n
€
Integral Test.
€
∫
∞
1
f (x)dx is easily evaluated, use the
The Test for Divergence
∞
If lim an does not exist of if lim an ≠ 0 , then the series
n →∞
n →∞
∑a
n
is divergent.
n=1
The Integral Test
Suppose f is a continuous,
positive, decreasing function on [1,∞) and let f (n) = an , then
€
€
€
i) If
∫
ii) If
∞
∞
f (x)dx is convergent, then
1
∫
is convergent.
∑a
n
n=1
∞
∞
f (x)dx is divergent, then
1
∑a
n
is divergent.
n=1
The Comparison Test
Suppose that ∑ an and
∑b
n
are series with positive terms and an ≤ bn (eventually), then
∑ b is convergent then ∑ a is convergent.
ii) If€ ∑ a is divergent
then ∑ b is divergent.
€
i) If
n
n
n
n
€
The
Test
€ Limit Comparison €
an
=c
€ Suppose ∑ an and€∑ bn are series with positive terms. If lim
n →∞ b
n
where c is a finite number and c > 0, then the series either both converge or both diverge.
The €
Alternating
€ Series Test
Suppose we have a series
i) lim bn = 0
n →∞
∑a
n
€ n
n−1
and either an = (−1) bn or an = (−1) bn where bn > 0 . If
and
ii) {bn } is a decreasing
sequence€
(eventually) i.e.€bn +1 < bn
€
then the series is convergent.
€
€The Ratio Test
€
€
an+1
n→∞ a
n
i) If L < 1, the series is absolutely convergent (and hence convergent).
ii) If L > 1 or if L = ∞ , the series is divergent.
iii) If L = 1, the Ratio Test is inconclusive.
Let
∑a
n
be a series and suppose L = lim
The Root Test
Let ∑ an be a series and suppose L = lim n an = lim an
n→∞
1
n
n→∞
i) If L < 1, the series is absolutely convergent (and hence convergent).
ii) If L > 1 or if L = ∞ , the series is divergent.
iii) If L = 1, the Root Test is inconclusive.