4. Test for Divergence
lim
n→∞
an ̸= 0 =⇒
∞
X
n=1
an diverges
▶ The first test you might consider doing is the test for
divergence, as it is the simplest.
▶ If lim
n→∞
an = 0 then test for divergence tells us nothing at all so
be careful not to make that mistake.
5. Test for Divergence
eg
∞
X
n=1
5n3 + n2 + 1
n2 + 3n
; Rational with higher degree in numerator
=⇒ the numerator dominates so lim
n→∞
an = ∞
eg
∞
X
n=1
5n2 + n2 + 1
n2 + 2n + 1
; Rational with same degree in numerator
and denominator =⇒ an approaches ratio of leading
coefficients lim
n→∞
an = 5
eg
∞
X
n=1
(−1)n
; an oscillates between two values, never getting
closer to either lim
n→∞
an DNE
7. Geometric Series
If we can write the series in the form
∞
X
n=1
arn−1
then the series converges iff |r| < 1.
▶ If it converges, then it converges to
a
1 − r
▶ Sometimes we need to manipulate the expression an in order
to get this form.
eg
∞
X
n=1
5n+1
7−n
=
∞
X
n=1
5
5
7
n
=
∞
X
n=1
25
7
5
7
n−1
=
25
2
8. Comparison Tests
▶ Use comparison Test:
▶ If
P
an is similar to a p-series.
▶ If an is a rational function, or an algebraic function (involving
roots of polynomials), compare to a p-series. Chose bn by
keeping the numerator of an and just the highest power in the
denominator of an.
eg To test
P n5
+3n2
n7+4
, compare to
P n5
+3n2
n7 =
P 1
n2 + 3
n5
▶ If
P
an is similar to a geometric series.
▶ Proceed similarly,
eg To test
P 5n
7n+4n , compare to the geo series
P 5n
7n
9. Comparison Tests
▶ Even though the comparison test is only for series with positive
terms, we can still use it on
P
|an| to test for absolute
convergence. If it is absolutely convergent, then it is convergent. If
it is not absolutely convergent, it could still be convergent so
consider a different test.
▶ When doing comparison:
▶ First determine series
P
bn what to compare to.
a) Does
P
bn converge where an ≤ bn for each n?
b) Does
P
bn diverge where bn ≤ an for each n?
▶ If either a) or b) is true, direct comparison works.
▶ If neither is true then do limit comparison.
11. Example 1
2/2
▶ Test for divergence won’t work since lim
n→∞
1
√
n2 + 1
= 0.
▶ Looks like a p-Series, so use a comparison test.
▶ Highest power term in denominator is
√
n2 so compare to
P 1
√
n2
=
P 1
n , which diverges.
▶ Since
n2
+ 1 ≥ n2
=⇒
p
n2 + 1 ≥
√
n2 = n =⇒
1
√
n2 + 1
≤
1
n
,
we can’t use direct comparison. Try limit comparison:
lim
n→∞
1
√
n2+1
1
n
= lim
n→∞
n
√
n2 + 1
= 1
So both series diverge.
12. Alternating Series Test
▶ Use AST whenever you have a series of the form
X
(−1)n−1
bn or
X
(−1)n
bn bn 0
▶ Note: If
∞
X
n=1
bn converges then the given series is absolutely
convergent and therefore convergent.
13. Ratio Test
▶ Use ratio test if
▶
P
an involves factorials.
▶
P
an involves multiple products including constant to power n.
▶ Don’t use for p−series (or any rational or algebraic functions on n),
you will always get L = 1.
14. Root Test
▶ Use root test if
▶ an is of the form (bn)n
▶ If root test is inconclusive then ratio test will also be inconclusive.
15. Integral Test
▶ Use integral test if
▶ an = f (n) where
R ∞
1
f (x)dx is easily computed and f is positive,
continuous, and decreasing (or eventually decreasing.)
17. Example 2
1/2
Determine which test or tests to use for convergence of the
following series. Then test it.
∞
X
n=1
n − 1
2n + 1
18. Example 2
2/2
▶ n−1
2n+1 is improper rational and
lim
n→∞
n − 1
2n + 1
=
1
2
so the series diverges by the test for divergence.
19. Example 3
1/3
Determine which test or tests to use for convergence of the
following series. Then test it.
∞
X
n=1
ne−n2
20. Example 3
2/3
▶ an = f (n) where f (x) = x
ex2 .
▶ f is clearly positive and continuous for x 0.
▶ f is decreasing:
f ′
(x) =
ex2
− x(2x)ex2
e2x2
=
ex2
(1 − 2x2)
e2x2
=
1 − 2x2
ex2 0 ⇐⇒
1 − 2x2
0 ⇐⇒
1
√
2
x ✓
▶ Let’s use the integral test.
21. Example 3
3/3
▶ Now computing the improper integral we use a simple
u-substitution:
Z ∞
1
xe−x2
dx = lim
t→∞
Z t
1
xe−x2
dx
= lim
t→∞
−
1
2
Z −t2
−1
eu
du [u = −x2
du = −2xdx]
= −
1
2
lim
t→∞
1
et2 −
1
e
=
1
2e
▶ Since the integral converges, the series
P
ne−n2
converges as well.
22. Example 4
1/2
Determine which test or tests to use for convergence of the
following series. Then test it.
∞
X
k=1
2k
k!
23. Example 4
2/2
▶ an = 2k
k! has a constant raised to n as well as a factorial, so let’s use
ratio test.
lim
k→∞
37. ≤
1
1 + 2n
and since
P 1
1+2n is similar to the geo series
P 1
2n , we use a
comparison test.
▶
P 1
2n converges and
1
1 + 2n
≤
1
2n
therefore
P 1
1+2n converges by direct comparison.
▶ Now using direct comparison again,
P
|an| =
P
