1.
Infinite Sequences and Series
2.Series
Series
What do we mean when we express a number as an infinite
decimal? For instance, what does it mean to write
= 3.14159 26535 89793 23846 26433 83279 50288 . . .
The convention behind our decimal notation is that any
number can be written as an infinite sum. Here it means that
where the three dots (. . .) indicate that the sum continues
forever, and the more terms we add, the closer we get to the
actual value of .
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Series
In general, if we try to add the terms of an infinite sequence
we get an expression of the form
a1 + a2 + a3+ . . . + an + . . .
which is called an infinite series (or just a series) and is
denoted, for short, by the symbol
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Series
It would be impossible to find a finite sum for the series
1 + 2 + 3 + 4 + 5 + ...+ n + ...
because if we start adding the terms we get the cumulative
sums 1, 3, 6, 10, 15, 21, . . . and, after the nth term, we get
n(n + 1)/2, which becomes very large as n increases.
However, if we start to add the terms of the series
we get
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Series
The table shows that as we add more and more terms,
these partial sums become closer and closer to 1.
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Series
In fact, by adding sufficiently many terms of the series we
can make the partial sums as close as we like to 1.
So it seems reasonable to say that the sum of this infinite
series is 1 and to write
We use a similar idea to determine whether or not a general
series (1) has a sum.
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Series
We consider the partial sums
s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
s4 = a1 + a2 + a3 + a4
and, in general,
sn = a1 + a2 + a3 + . . . + an =
These partial sums form a new sequence {sn}, which may or
may not have a limit.
8
Series
If limn → sn = s exists (as a finite number), then, as in the
preceding example, we call it the sum of the infinite series
an.
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Series
Thus the sum of a series is the limit of the sequence of
partial sums.
So when we write we mean that by adding
sufficiently many terms of the series we can get as close as
we like to the number s.
Notice that
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Example 1
An important example of an infinite series is the geometric
series
a + ar + ar2 + ar3 + . . . + ar n–1 + . . . = a0
Each term is obtained from the preceding one by multiplying
it by the common ratio r.
If r = 1, then sn = a + a + . . . + a = na →
Since limn → sn doesn’t exist, the geometric series diverges
in this case.
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Example 1 cont’d
If r 1, we have
sn = a + ar + ar2 + . . . + ar n-1
and rsn = ar + ar2 + . . . + ar n-1 + ar n
Subtracting these equations, we get
sn – rsn = a – ar n
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Example 1 cont’d
If –1< r < 1, we know that as r n → 0 as n → ,
so
Thus when | r| < 1 the geometric series is convergent and its
sum is a/(1 – r).
If r –1 or r > 1, the sequence {r n} is divergent and so, by
Equation 3, limn → sn does not exist.
Therefore the geometric series diverges in those cases.
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Series
We summarize the results of Example 1 as follows.
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Example 7
Show that the harmonic series
is divergent.
Solution:
For this particular series it’s convenient to consider the
partial sums s2, s4, s8, s16, s32, . . . and show that they
become large.
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Example 7 – Solution cont’d
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Example 7 – Solution cont’d
Similarly, and in general
This shows that and so {sn} is divergent.
Therefore the harmonic series diverges.
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Series
The converse of Theorem 6 is not true in general.
If limn→ an = 0, we cannot conclude that an is convergent.
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Series
The Test for Divergence follows from Theorem 6 because, if
the series is not divergent, then it is convergent, and so
limn → an = 0.
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Series
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