Integral calculus—Integral calculus (logically) deals with integration and its appli-
cation to solve differential equations; it is also used to determine areas and volumes.
Other various analyses—Other parts of calculus entail various types of analyses,
such as vector, tensor, and complex analyses, and differential geometry.
Also remember that the term “calculus” is a generic name for any area of mathe-
matics dealing with calculation; thus, arithmetic could be called the “calculus of num-
bers.” It is also why there are such terms as imaginary calculus—or a method of look-
ing at the relationships between real or imaginary quantities using imaginary symbols
and quantities in algebra—that do notmean the type of calculus discussed elsewhere
in this chapter.
SEQUENCES AND SERIES
What is a sequence?
A sequence is defined as a set of real numbers with a natural order. A sequence is usu-
ally included in brackets ({}), with the terms,or parts of a sequence, separated by com-
mas. For example, if a scientist collects weather data every day for many days, the first
day of collecting can be written as x 1 data; then x 2 for the second day, and so on until
xn, in which nis the eventual number of days. This can be written as {x 1 , x 2 , ... xn} n≥ 1.
In general, the sequence of numbers in which xnis the nthnumber is written using
the following notation: {xn}n≥ 1.
A sequence can get larger or smaller. For example, in the sequence for {2n}n≥ 1 , the
solution is 2 ≤ 4 ≤ 8 ≤ 16 ≤32, and so on, with the numbers getting larger. Whereas,
for {1/n}n≥ 1 , the sequence becomes 1 ≥1/2 ≥1/3 ≥1/4 ≥1/5, and so on, with the num-
bers getting progressively smaller. This does not mean that sequences only get pro-
gressively larger and smaller; certain solutions for sequences include a mix of the two.
What is the range of a sequence?
The range of a sequence is merely a set that defines the sequence. The range is usually
represented by the set {x 1 }, {x 2 }, {x 3 }, and so on; it is also written as {xn; n1, 2, 3,
...}. For example, in the question above, the data for each day that the scientist col-
lects from his weather experiment is the range. Another example is the range of the
sequence {(1)n}n≥ 1 : It is the two-element set {1, 1}.
When is a sequence monotonic?
A sequence is called monotonic if one of the following properties hold: In the
sequence {xn}n≥ 1 , it is increasing if and only if xn< xn 1 for any n≥1, or it is decreas-
ing if and only if xn> xn 1 for any n≥1. 213