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数学专业英语SequencesandSeries.pdf
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数学专业英语-Sequences and Series
Series are a natural continuation of our study of functions. In the previous cha
pter we found how
to approximate our elementary functions by polynomials, with a certain error te
rm. Conversely, one can define arbitrary functions by giving a series for them.
We shall see how in the sections below.
In practice, very few tests are used to determine convergence of series. Esse
ntially, the comparision test is the most frequent. Furthermore, the most import
ant series are those which converge absolutely. Thus we shall put greater emp
hasis on these.
Convergent Series
Suppose that we are given a sequcnce of numbers
a
1
,a
2
,a
3
…
i.e. we are given a number a
n
, for each integer n>1.We form the sums
S
n
=a
1
+a
2
+…+a
n
It would be meaningless to form an infinite sum
a
1
+a
2
+a
3
+…
because we do not know how to add infinitely many numbers. However, if ou
r sums S
n
approach a limit as n becomes large, then we say that the sum of
our sequence converges, and we now define its sum to be that limit.
The symbols
∑
a=1
∞
a
n
will be called a series. We shall say that the series converges if the sums app
roach a limit as n becomes large. Otherwise, we say that it does not converge,
or diverges. If the seriers converges, we say that the value of the series is
∑
a=1
∞
=lim
a
→∞
S
n
=lim
a
→∞
(a
1
+a
2
+…+a
n
)
In view of the fact that the limit of a sum is the sum of the limits, and other
standard properties of limits, we get:
THEOREM 1. Let{ a
n
}and { b
n
}(n=1,2,…)
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