INFINITE SERIES (Chapter 11)
General facts:
A
sequence
is nothing but an infinite list. A
series
is a
sum
of infinitely many numbers.
An infinite series may either
converge
(i.e. reach a finite total) or
diverge
(i.e. not reach a finite total).
A famous example of a
convergent
series is any geometric series with

r

<
1 (e.g.
r
=
1
2
):
∞
X
n
=1
1
2
n
=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+
· · ·
converges
A famous example of a
divergent
series is the harmonic series:
∞
X
n
=1
1
n
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
· · ·
diverges
These are two very famous series, so remember them.
(To
prove
that these two series converge and
diverge respectively, you have to do a bit of algebra.)
So, if
a
n
=
1
2
n
, the series
X
a
n
converges. If
a
n
=
1
n
, the series
X
a
n
diverges.
VERY IMPORTANT:
The above shows that if
a
n
approaches 0, this does
NOT GUARANTEE
that the series
X
a
n
converges!
•
If
a
n
does NOT approach 0, then the series
X
a
n
MUST DIVERGE.
•
If
a
n
DOES approach 0, then the series
X
a
n
MIGHT CONVERGE or MIGHT DIVERGE.
More work is needed to decide which! You’re not done!
Rough idea: If
a
n
approaches 0 “fast enough”, the series
X
a
n
converges. If
a
n
doesn’t approach 0 “fast
enough”, the series
X
a
n
diverges.
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In the
MOST COMMON TYPE OF EXAM PROBLEM
on infinite series, you
will be given an infinite series
X
a
n
where the formula for
a
n
might be complicated.
You will have to decide (with proof) whether the series converges or diverges.
Two famous series:
GEOMETRIC SERIES.
Any infinite series of the form
a
+
ar
+
ar
2
+
ar
3
+
ar
4
+
· · ·
=
∞
X
n
=0
ar
n
=
∞
X
n
=1
ar
n

1
is convergent if

r

<
1, and is divergent if

r
 ≥
1. If it converges, its sum is
a
1

r
.
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 Spring '08
 wang
 Infinite Series, Mathematical Series

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