11.2 Series
Adding the terms in an infinite sequence creates an
infinite series.
12
1
nn
n
aaa
a
a
∞
=
=++
+
+
∑∑
LL
The question is: Can we determine if the series has a sum?
It’s not possible to add an infinite number of terms, but it is possible to consider
partial sums
to determine whether a series has a sum. It is important that you understand the difference
between a sequence and a series. A sequence is a list of numbers and a series is the sum of all
those numbers.
Your calculator can give you a sum of a finite number of terms in a sequence. Let’s look at the
series
1
2
3
n
n
∞
=
∑
.
The first window shows the sum of the first 10 terms in the sequence. The
second window shows the sum of the first 20 terms. Do you think the series has a sum?
The two sums we looked at above are partial sums. The partial sums themselves form a
sequence, which may or may not have a limit.
Look at the definition on page 705.
1
n
n
a
∞
=
=
∑
∞
s
means that by taking
n
large enough, we can get as close as we want to the number
s.
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 Fall '08
 Estrada
 Infinite Series, lim, partial sums

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