470
8.3. SERIES OF REAL NUMBERS
number e as the infinite sum
e =
∞
k=0
1
k!
.
(8.13)
This sum is an example of a series (or an infinite series). Note that the series (8.13) is the
sum of the terms of the (infinite) sequence
1
n!
. In general, we use the following notation
and terminology.
Definition 8.3. An infinite series of real numbers is the sum of the entries in an infinite
sequence of real numbers. In other words, an infinite series is sum of the form
a 1 + a 2 + · · · + a n + · · · =
∞
k=1
a k ,
where a 1 , a 2 , . . ., are real numbers.
We will normally use summation notation to identify a series. If the series adds the
entries of a sequence {a n } n≥1 , then we will write the series as
k ≥1
a k
or
∞
k=1
a k .
Note well: each of these notations is simply shorthand for the infinite sum a 1 + a 2 + · · · +
a n + · · · .
Is it even possible to sum an infinite list of numbers? This question is one whose
answer shouldn’t come as a surprise. After all, we have used the definite integral to add up
continuous (infinite) collections of numbers, so summing the entries of a sequence might
be even easier. Moreover, we have already examined the special case of geometric series
in the previous section. The next activity provides some more insight into how we make
sense of the process of summing an infinite list of numbers.
Activity 8.7.
Consider the series
∞
k=1
1
k 2 .
While it is physically impossible to add an infinite collection of numbers, we can,
of course, add any finite collection of them. In what follows, we investigate how
understanding how to find the nth partial sum (that is, the sum of the first n terms)
enables us to make sense of the infinite sum.
8.3. SERIES OF REAL NUMBERS
number e as the infinite sum
e =
∞
k=0
1
k!
.
(8.13)
This sum is an example of a series (or an infinite series). Note that the series (8.13) is the
sum of the terms of the (infinite) sequence
1
n!
. In general, we use the following notation
and terminology.
Definition 8.3. An infinite series of real numbers is the sum of the entries in an infinite
sequence of real numbers. In other words, an infinite series is sum of the form
a 1 + a 2 + · · · + a n + · · · =
∞
k=1
a k ,
where a 1 , a 2 , . . ., are real numbers.
We will normally use summation notation to identify a series. If the series adds the
entries of a sequence {a n } n≥1 , then we will write the series as
k ≥1
a k
or
∞
k=1
a k .
Note well: each of these notations is simply shorthand for the infinite sum a 1 + a 2 + · · · +
a n + · · · .
Is it even possible to sum an infinite list of numbers? This question is one whose
answer shouldn’t come as a surprise. After all, we have used the definite integral to add up
continuous (infinite) collections of numbers, so summing the entries of a sequence might
be even easier. Moreover, we have already examined the special case of geometric series
in the previous section. The next activity provides some more insight into how we make
sense of the process of summing an infinite list of numbers.
Activity 8.7.
Consider the series
∞
k=1
1
k 2 .
While it is physically impossible to add an infinite collection of numbers, we can,
of course, add any finite collection of them. In what follows, we investigate how
understanding how to find the nth partial sum (that is, the sum of the first n terms)
enables us to make sense of the infinite sum.