The Handy Math Answer Book

For example, in order to check that the sequence {2n}n≥ 1 is increasing: Let n≥1;

that gives 2n^1  2 n2. Because 2 is greater than 1, which means that 1  2 n< 2  2 n;

thus 2n< 2n^1 , which shows the sequence is increasing.

What are the bounds of a sequence?

Once again, take the sequence {xn}n≥ 1. This sequence is bounded aboveif and only if

there is a number Msuch that xn≤M(the Mis called an upper-bound). In addition,

the sequence is bounded belowif and only if there is a number msuch that xn≥m

(the mis called a lower-bound). For example, the sequence {2n}n≥ 1 , is bounded below

by 0 because it is positive, but not bounded above.

The sequence is usually said to be merely bounded(or “bd” for short) if both of

the properties (upper- and lower-bound) hold. For example, the harmonic sequence

{1, 1/2, 1/3, 1/4 ...} is considered bounded because no term is greater than 1 or less

than 0; thus, the upper- and lower-bounds, respectively, apply.

What is a limit of a sequence?

The limit of a sequence is simply the number that represents a kind of equilibrium

reached in the sequence. It is also phrased “approaches as closely as possible.” (Limit

is also a term used in calculus in relation to a function; see elsewhere in this chapter.)

What are the concepts of convergentand divergent sequences?

Convergent and divergent sequences are based on the limit of a sequence. A conver-

gent sequence, the one most commonly worked on in calculus, means that one math-

ematical sequence gets close to another and eventually approaches a limit (conver-

gence can also apply to curves, functions, or series). This is seen visually when a curve

approaches the xor yaxes but does not quite reach it. For example, take the sequence

214 of numbers used above, or {xn}n≥ 1. Often the numbers come closer and closer to a

How can a calculator determine the limit of a sequence?

T

here is an excellent way to understand the limit of a sequence by using a cal-

culator. In scientific calculators that include geometric functions (such as

cosine, sine, and tangent), a limit is easy to see: Find x 1 cos (1), then x 2 cos

(x 1 ), and so on. Just put the calculator in the “radian” mode, enter “1,” and then

hit the “cosine” key repeatedly. The number will start at “0.540302305,” then

change to “0.857553215,” and keep changing. As it approaches about the twenti-

eth change, the amount gets closer and closer to a number that begins as “0.73

...”. This indicates the limit of the sequence is being reached.