Guidelines for Analyzing Infinite Series
Math 143
Kaul
1. General Guidelines
• The only way to become proficient at analyzing series is through
LOTS OF PRACTICE.
• Know the statements of all definitions, theorems and tests.
• Remember to distinguish between a series and a sequence.
• The format for solutions used in class (which is identical to the format found in the text)
will be expected. This means that, for each test used, you must clearly show
(i) that it is applicable (eg., if a particular test requires an to be decreasing, then you
must show that an is decreasing, either directly or by computing the appropriate
derivative);
(ii) the work involved in applying the test.
2. The Tests...and when to use them
Remember: there is often more than one correct way to analyze an infinite series.
(a) p-series
Deciding whether a p-series is convergent/divergent is straighforward. These series are
often used for comparison in the CT and LCT.
∞
X
1
n=k
np
(
is
convergent if p > 1
divergent if p ≤ 1
(b) Geometric Series Test
Also fairly straighforward, though you might have to manipulate the sequence before
applying the formula.
∞
X
n=k
(
r
n
k
r
if −1 < r < 1
= 1−r
is divergent if
|r| ≥ 1
(c) Test for Divergence (TFD)
As the name suggests, this test is used to show divergence of a series. It cannot be
used to show that a series converges. If lim an 6= 0, then
n→∞
∞
X
an is divergent.
n=k
(d) Comparison Tests
If
∞
X
an “resembles” a p-series or geometric series then the CT or LCT usually works.
n=k
Assume an , bn > 0. If
1
CT:
an ≤ bn and
an ≥ bn and
∞
X
n=k
∞
X
bn is convergent, then
bn is divergent, then
n=k
∞
X
an is convergent.
n=k
∞
X
an is divergent.
n=k
an
< ∞, then both series either converge or diverge.
bn
Choosing bn is based on
LCT: 0 < lim
n→∞
i. whether you suspect convergence or divergence of the original series;
ii. ignoring “non-dominating” terms of an .
When an has non-positive terms, CT/LCT can be used when checking for absolute
convergence.
(e) Alternating Series Test
Apply AST to any series with decreasing terms that can be expressed as
∞
X
(−1)n an
n=k
with an > 0.
If lim an = 0, then the series is convergent.
n→∞
Note: If lim an 6= 0, then lim (−1)n an 6= 0 as well and the series diverges by the TFD,
n→∞
n→∞
NOT the AST.
(f) Root and Ratio Tests
One (or both) of these tests usually works when
∞
X
an involves factorials and/or expo-
n=k
nentiation. They will usually fail if the series is “p-like.”
i. (Ratio) If


<1

an+1 is
>
1,
=∞
lim n→∞ an 

=1
series is absolutely convergent
series is divergent
no information
ii. (Root) If
lim
n→∞
q
n
|an | is



<1
series is absolutely convergent
> 1, = ∞
series is divergent


=1
no information
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(g) Telescoping Series
Somewhat obscure, but a good measure of whether you know know the definition of
convergence of a series. It is also one of the instances when we can actually find the
actual sum of the infinite series (assuming it converges). Recognizing that a series is
telescoping means that you have expressed
N
X
SN =
an
n=k
in a nice compact form (usually via partial fractions). Once you have done so, you simply
compute S = lim SN (or, in the case that the series diverges, show that the limit does
N →∞
not exist).
(h) The Integral Test
If f (x), is continuous, positive, decreasing and f (n) = an , then
i.
ii.
∞
X
n=k
∞
X
an converges ⇔
Z ∞
f (x) dx converges;
k
an diverges ⇔
Z ∞
f (x) dx diverges.
k
n=k
3. Approximation
We have two methods of approximating an infinite series by a finite sum (with error < ).
Both methods involve finding the (minimal) N such that |S − SN | ≡ RN < .
(a) (IT)
Assume that
∞
X
an is convergent by the IT. Then
n=k
RN ≤
Z ∞
f (x) dx
N
(so find N such that the integral is less than ).
(b) (AST) Assume that
∞
X
(−1)n an is convergent by the AST. Then RN ≤ aN +1 (so find
n=k
N such that the (N + 1)−th term is less than ).
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