Lecture Notes - Series

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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, " " " "     â œ " # % ) "'

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 +