Series. Lecture Notes. A. : series is an infinite sum of numbers. " " " " #. %. ) "' .
... By definition, the sum of the series is the limit of this sequence: DEFINITION ...
Series Lecture Notes
A series is an infinite sum of numbers: " " " " â # % ) "' The individual numbers are called the terms of the series. In the above series, the first term is "Î#, the second term is "Î%, and so on. The 8th term is "Î#8 : " " " " " â 8 â # % ) "' # We can express an infinite series using summation notation. For example, the above series would be written as follows: " _
8œ"
" #8
The " symbol means "sum". The 8 œ " on the bottom refers to the fact that the first term is
8 œ " (some series start with 8 œ ! or 8 œ #), and the _ on the top indicates that the series continues indefinitely.
The Sum of a Series Many series add up to an infinite amount. For example: "" œ " " " " " â œ _ _
8œ"
"8 œ " # $ % & â œ _ _
8œ"
However, if the terms of a series become smaller and smaller, it is possible for the series to have a finite sum. The basic example is the series " _
8œ"
which sums to ".
" #8
œ
" " " " " â # % ) "' $#
But what exactly does it mean to find the sum of an infinite series? Given a series "+8 œ +" +# +$ â, _
8œ"
a partial sum is the result of adding together the first few terms. For example, the third partial sum is +" +# +$ , and the hundredth partial sum is +" +# â +"!! . The partial sums form a sequence: =" œ +" =# œ +" +# =$ œ +" +# +$ =% œ +" +# +$ +% ã By definition, the sum of the series is the limit of this sequence:
DEFINITION (SUM OF A SERIES) The sum of the series " +8 is the limit of the partial sums: _
8œ"
" +8 œ _
8œ"
lim =8 œ
8Ä_
lim a+" â +8 b
8Ä_
EXAMPLE 1 Consider the series " _
8œ"
" #8
œ
" " " " â # % ) "'
The first few partial sums of this series are listed below: =" œ
" #
=# œ
" " œ # %
$ %
=$ œ
" " " œ # % )
=% œ
" " " " œ # % ) "'
( ) "& "'
As you can see, the partial sums are converging to ". Therefore, according to our definition of
the sum of an infinite series, " " " " ⠜ " # % ) "'
è
EXAMPLE 2 Find the sum of the series: !Þ$ !Þ!$ !Þ!!$ !Þ!!!$ â SOLUTION
Here are the first few partial sums: =" œ
!Þ$
=# œ !Þ$ !Þ!$ œ
!Þ$$
=$ œ !Þ$ !Þ!$ !Þ!!$ œ
!Þ$$$
=% œ !Þ$ !Þ!$ !Þ!!$ !Þ!!!$ œ
!Þ$$$$
As you can see, the partial sums are converging to the repeating decimal !Þ$$$$á , which is equal to "Î$. è We use the same terminology for sums of series that we do for limits of sequences and for improper integrals:
CONVERGENCE AND DIVERGENCE
We say the series ! +8 converges if its sum is a real number.
If the sum is infinite or does not exist, then the series ! +8 diverges. Keep in mind that a series can only converge if its terms get smaller and smaller. That is, the
sum " +8 can only be finite if lim +8 œ !. _
8œ"
8Ä_
DO THE TERMS APPROACH ZERO? 1. If lim +8 Á !, then the series " +8 diverges. _
8Ä_
8œ"
2. If lim +8 œ !, then the series " +8 may converge, or it may diverge. _
8Ä_
8œ"
EXAMPLE 3 Find the sum of the series: " # $ % & â # $ % & ' Since the individual terms of the series are getting closer and closer to ", the sum of the series is infinite (for the same reason that " " " â œ _). è SOLUTION
EXAMPLE 4 Determine whether the series " _
8œ"
SOLUTION
8 converges or diverges. #8 "
8 " œ Á !, this series diverges to _. 8Ä_ #8 " #
Since lim
è
By the way, it is quite possible for the sum of a series to be infinite even if the terms get smaller and smaller. For example, even though the terms of the series "
" " " " â # $ % &
become smaller and smaller, the sum of this series is infinite! This series is important enough to have its own name: the harmonic series (named for the frequencies of harmonic overtones in music). You should always remember that the harmonic series diverges.
Geometric Series A geometric series is the sum of the terms of a geometric sequence. For example, the series "
" " " " â # % ) "'
is geometric, with a common ratio of "Î# (i.e. each term is "Î# times the previous term). Here are several more examples: "# % "
% % % â $ * #(
# % ) â & #& "#&
$ #
% ) "' â $ * #(
(common ratio of "Î$) (common ratio of #Î&) (common ratio or #Î$)
In our study of geometric sequences, we learned that the formula for the 8th term of a geometric sequence has the form +