number we’ll call L; written in calculus, xnL. If the numbers do come closer, the
sequence is said to be convergentand has a limit equal to L. Conversely, if the
sequence is not convergent, it is called divergent.
Most mathematicians and scientists are not only interested in how a sequence
converges (or diverges), but also how fast it converges, which is called the speed of
convergence. There are several basic properties of the limits of a sequence, including
that the limit of a convergent sequence is unique, every convergent sequence is
bounded, and any bounded increasing or decreasing sequence is convergent.
How are limitswritten in terms of convergent and divergent sequences?
Limits of convergent and divergent sequences are written as follows (the figure eight
on its side is the symbol for infinity):
or xn$Lwhen n$.What is a series?
A series is closely related to the sum of numbers. It is actually used to help add num-
bers; therefore, in a sequence it can be the indicated sum of that sequence. In general,
the idea is to start with a number, then do something to that number to get the next
number, then do the same to that number to get the next number, and so on. For
example, a finite serieswith six terms is 2 4 6 8 10 12, in which 2 is added
to each number to get the next number. An example of an infinite seriesis one with
the notation 1/2n, with n≥1, or 1/2 1/4 1/8 ... (and so on).
To see a series written in notation, if set {xn} is a sequence of numbers being added,
and set s 1 x 1 , then s 2 x 1 x 2 ; s 3 x 1 x 2 x 3 ; and so on. And for n≥1, a new
sequence is made, {sn}, called the sequence of partial sums,or snx 1 x 2 ... xn.
What is the notationcommonly used for series and sequences?
When looking at a sequence and series, we need to distinguish between the ones we
want to add, and the ones we do not want to add. The common symbol used for addi-
tion with a sequence or a series is the summation symbol, seen as the symbol . When
discussing a series, this symbol is used to mean the sum of numbers. For example, for
the series called xn:
Also note that for any given series xn, we have to also remember to associate it
with the sequence of partial sums (snx 1 x 2 ... xn).
xn
n$ 1!
nlim" 3 xLn=