1 Sequences, Series, how to decide if a series in convergent

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1
Sequences, Series, how to decide if a series
in convergent
This is a summary of results about series used in this course. I have included
sketches of some of the proofs, but read these only if you are curious; they
are not part of the course. Almost all calculus texts cover infinite series so
you can easily look up detailed arguments if you want to.
Sequences. A sequence is a list (a1 , a2 , . . . ) of numbers. The numbers in
the sequence are called terms. The label on the term specifies the order: a1 is
the first term, a2 is the second, and so on. The · · · means that the sequence
has no last element and is therefore an infinite sequence. Sequences can be
finite, but then you write the last term. For example (a1 , . . . , aN ) is a finite
sequence with N terms. The terms of a sequence are allowed to be the same,
for example (1, 1, 2, 2) is a sequence.
A set of numbers is written {a1 , a2 , . . . , aN }. Unlike a sequence the numbers in a set must all be different, and if you put the numbers in a different
order such as aN , aN −1 , . . . , a1 , then the set {aN , aN −1 , . . . , a1 } is not a different set:
{aN , aN −1 , . . . , a1 } = {a1 , a2 , . . . , aN }.
(1)
Thus there is no order on the numbers in the set; a1 is not the first element
in the set because the phrase “first element” has no meaning for a set. Notice
also that we do not use the word term, but instead refer to a1 as an element.
The different language helps to train us not to confuse sequences with sets.
Finite Series. Given a sequence (a1 , a2 , . . . aN ) form the sum a1 +a2 +· · ·+
aN . This is called a finite series because it is the sum of a finite sequence
(not
PNbecause it equals a finite number). Another notation for the same sum
is i=1 ai , but if you use this then you have to realise that you can equally
P
well use N
j=1 aj : changing i to j adds up the same numbers a1 , a2 , · · · , aN
in the same order. Since we can change the name of the index i without
ruining any equations in which this sum occurs we call i a dummy index.
The numbers a1 , a2 , . . . , aN are called terms in the series a1 + a2 + . . . + aN .
1
The most important finite series is 1 + r + r2 + · · · + rN , where r is a given
real or complex number. This is called a finite geometric series with ratio r.
Exercise Show that 1 + r + r2 + · · · + rN =
1−rN +1
.
1−r
Infinite Series. An infinite series is an expression
the form a1 +a2 +. . . .
Pof
∞
An alternative notation for the same expression is Pn=1 an . Thus, given any
sequence (a1 , a2 , . . . ) there is the associated series ∞
n=1 an . Many egregious
errors come confusing series with sequence, so train yourself to always ask
“is this a statement about a series or is it a statement about a sequence?”
The series a1 +a2 +· · · is called an infinite series because it is formed from
an infinite sequence. It has nothing to do with whether the sum a1 + a2 + · · ·
“adds up to an infinite number”.
We will now explain what it means for a series a1 + a2 + . . . to converge to
a number L. We do it in terms of things we have already defined as follows.
Form the sequence
(a1 , a1 + a2 , a1 + a2 + a3 , . . . )
(2)
which you can also write as
1
X
i=1
ai ,
2
X
ai ,
i=1
3
X
!
ai , . . .
.
(3)
i=1
The terms in this sequence are called partial sums of the series a1 + a2 + . . .
and the sequence itself is called the sequence of partial sums.
Definition 1.1 The series a1 + a2 + . . . is said to be convergent to L if the
sequence of partial sums has a limit and this limit is L. More concisely, if
lim
N →∞
N
X
ai = L.
(4)
i=1
P
When a1 +a2 +. . . is convergent to L we write a1 +a2 +· · · = L or ∞
n=1 an =
L.
When the sequence has a limit but we do not want to be specific about
what the limit is we say that a1 + a2 + · · · is convergent. A series which is
not convergent is said to be divergent.
An example of a divergent series is 1 − 1 + 1 − 1 + · · · .
2
Exercise. Show, using the definitions, that the geometric series 1 + r +
1
provided r ∈ (−1, 1) and otherwise is divergent.
r2 + · · · is convergent to 1−r
Manipulating infinite series. Here are some useful formulas that are valid
when the series are convergent.
∞
X
ai = a1 + a2 + · · · + an +
i=1
∞
X
c
i=1
∞
X
∞
X
ai
(5)
i=n+1
ai =
∞
X
cai
(6)
i=1
(ai + bi ) =
i=1
∞
X
ai +
i=1
∞
X
bi
(7)
i=1
These are quite easy to prove from our definition, but I omit the proofs.
The important thing to remember when you get confused by some statement
about infinite series is that they are not ordinary sums as finite series are,
they are limits of partial sums. Here is an example of how an infinite series
is not like a finite series:
Exercise. Start with the series
1+
−1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1
+ +
+ +
+ +
+ +
+ +
+ +
+ +· · · (8)
2 2 4 4 4 4 8 8 8 8 8 8 8 8
The terms alternate in sign, there are two halves, four quarters, eight 1/8’s,
etc. Observe that this series converges to 1. Now show that there is a way
to change the order of the terms in this series so that it converges to 2. Hint.
Put the sequence in an order that starts with
1+
1 1 1 −1 1 1 1 1 −1
+ + +
+ + + + +
+ ···
2 4 4
2
8 8 8 8
4
(9)
The rule to be followed is that every term in the old order must, sooner or
later, be included, so the terms I seem to be omitting are coming later than
I have written so far in this new order.
Series with non-negative terms. This exercise shows that the order
of terms in series with both positive and negative terms can affect what
the series converges to. However series where all terms have the same sign
3
will converge to the same limit regardless of how they are re-ordered. This
statement is part of the section called absolute convergence if you look up
series in a textbook.
We need simple ways to decide whether series converge or diverge. Lets
restrict our attention to series where all terms are the same sign; assume they
are all non-negative.
Theorem 1.2 For a series a1 + a2 + · · · with non-negative terms, if there is
a number M such that all partial sums are less than M , then a1 + a2 + · · ·
is convergent.
You may think this theorem is obvious once you understand what it says
but it is impossible to prove without defining the real numbers so actually
it is very deep. For example if, like the ancient Greeks, our number system
√
was just the rational numbers, and therefore did not include things like 2
then it would be false because we can make an √
infinite series
√ whose terms are
rational numbers which wants to converge to 2, but 2 will not be there
for the series to converge to! OK, so we put into our number system some
irrational numbers so that the series with rational terms will have something
to converge to, but now we can form series whose terms are irrational: how do
we know that these new series will not want some “hyper-irrational numbers”
between the ones we have just added in? It could go on forever, but it doesn’t.
Math 220 and 320 are about the foundations of analysis that lead up to proofs
of these hard theorems.
Consider the series
∞
X
1
i=1
i
=1+
1 1
+ + ···
2 3
(10)
Does it converge? What about the series
∞
X
1
1
1
= 1 + 2 + 2 + ···
2
i
2
3
i=1
(11)
These are called p series because they have the form
∞
X
1
,
p
i
i=1
with p = 1, 2, respectively.
4
(12)
Theorem 1.3
P∞
1
i=1 ip
converges for p > 1 and otherwise diverges.
In particular,
1 1
+ + ···
(13)
2 3
is divergent, a fact that many people find surprising. This series is called the
harmonic series. The proof of this theorem procedes by first proving that
1+
∞
X
1
,
p
i
i=1
is convergent iff and only if the integral
Z ∞
1
dx
xp
1
(14)
(15)
converges. This integral can be evaluated easily and one finds that it exists
if and only if p > 1, leading to the result claimed in the theorem. This
relation between convergence of series and convergence of integral is called
the integral test. Here is how the integral test is proved: think of the series as
the sum of areas of rectangles of width one and height 1i which are drawn side
by side under the graph of x1p . Then if the integral, which is the area under
1
, is finite, the sum must be smaller and so converges, by Theorem 1.2. If
xp
the integral is infinite then shift all the rectangles one unit to the right and
you will see that now the graph of x1p lies below the tops of the rectangles
and so the infinite area under x1p is smaller than the series, which is therefore
divergent.
5
Suppose that we have the series
∞
X
i=2
1
1
1
1
=
+
+
+ ···
i2 − i
22 − 2 32 − 3 42 − 4
(16)
Does it converge? Here is the theorem that helps us out. For two infinite
sequences (a1 , a2 , . . . ) and (b1 , b2 , . . . ) we make the definition,
Definition 1.4 ai ∼ bi if limi→∞
ai
bi
= 1.
Theorem 1.5 For any n, m ≥ 1, let an + an+1 + . . . and bm + bm+1 + . . .
be infinite
P∞let ai ∼ bi . THEN the
P series whose terms are nonnegative and
a
converges
if
and
only
if
the
series
series ∞
i=m bi converges.
i=n i
Take ai =
that
1
i2
and bi =
1
i2 −i
and n = m = 2 and apply the theorem to find
1
1
1
+
+
+ ···
22 − 2 32 − 3 42 − 4
(17)
converges.
P
Here are the ideas in the proof of the theorem. Whether a series ∞
i=n ai
converges is not affected by the value of nPbecause of (5) so below you will see
the
an argument about the convergence
of ∞
i=n ai and n will be chosen
Pin
P∞
∞
argument. Suppose that i=1 bi converges. We want to deduce that i=n ai
converges. Since ai ∼ bi we have ai /bi is close to one if i is sufficiently large.
In particular, close to one means that ai /bi ≤ 2. Sufficiently large means
that there is an integer n such that ai /bi ≤ 2 for all i ≥ n, which we write
as ai ≤ 2bi for all i ≥ n. Therefore, for any N ,
N
X
ai ≤
i=n
and so by theorem 1.2
we can also prove that
P∞
i=n
N
X
i=n
∞
X
2bi = 2
i=n
∞
X
bi
(18)
i=n
ai is convergent. By a variation on this theme
ai ≥
∞
X
1
i=n
∞
1X
bi =
bi
2
2 i=n
and this is used
to show the only if part of the claim: if
P∞
converge then i=1 ai does not converge.
6
(19)
P∞
i=1 bi
does not