Series Summary
Asequence is an ordered list of numbers: {an}={a1, a2, a3, . . .}, and a series is the
sum of those numbers:
∞
X
n=1
an=a1+a2+a3+· · · . In either case, we want to determine
if the sequence converges to a finite number or diverges and if the series converges to a
finite number or diverges. If the series converges, that means that a sum of infinitely many
numbers is equal to a finite number! If the sequence {an}diverges or converges to anything
other than 0, then the series Pandiverges. If the sequence {an}converges to 0, then the
series Panmay converge or may diverge.
For any given series Panthere are two associated sequences: the sequence of terms
{an}and the sequence of partial sums {sn}, where sn=a1+a2+. . . +an. If Pan=L,
then lim
n→∞
an= 0 (as stated above) and lim
n→∞
sn=L.
1 When can we calculate the sum of a series?
Unfortunately, we are unable to compute the exact sum of a series in most cases. However,
there are a few examples that can be computed.
Geometric Series For |r|<1, the series converges to a
1−r.
∞
X
n=1
arn−1For |r| ≥ 1, the series diverges.
Telescoping Series Also known as “canceling pairs”, subsequent pairs
∞
X
n=1
(bn−bn+c) of the series terms may cancel with each other.
2 Tests for determining if a series converges or diverges
In most cases, we will not be able to compute the exact sum of a series, but there are several
tests which allow us to at least determine if a series is convergent or divergent. In some cases
we can give approximations for the sum of a series as well.
Test for Divergence If lim
n→∞
an6= 0, then the series
∞
X
n=1
andiverges.
p-Series Test For p > 1, the series converges.
∞
X
n=1
1
npFor p≤1, the series diverges.
1
