limit of real number sequence

limit of real number sequence

An endless

real number

sequence

a

1

,

a

2

,

a

3

,

…

(1)

has the real number

A

as its

limit

, if the

distance

between

A

and

a

n

can be made smaller than an arbitrarily
small positive number

ε

by chosing the

n

of

a

n

sufficiently great, i.e. greater than a number

N

(the of which depends on the value of

ε

); accordingly

|

A

-

a

n

|

<

ε

when

n

>

N

.

Then we may denote

lim

n

→

∞

⁡

a

n

=

A

(2)

or equivalently

a

n

→

A

as

n

→

∞

.

(3)

Remark 1.

One should not think, that

a

n

=

A

when

n

=

∞

.  The symbol “

∞

” no number, one cannot set it for the value of

n

.  It’s only a question of allowing

n

to exceed any necessary
value.

Example 1.

Using the notation (2) we can write a result

lim

n

→

∞

⁡

2

⁢

n

n

+

1

=

2

.

It’s a question of that the real number sequence

2

2

,

4

3

,

6

4

,

…

has the limit value 2 (e.g. the nine hundred ninety-ninth member

1998

1000

=

1.998

is already “almost” 2!).  For justificating the result, let

ε

be an arbitrary positive number, as small as you want.  Then

|

2

-

2

⁢

n

n

+

1

|

=

|

2

⁢

n

+

2

n

+

1

-

2

⁢

n

n

+

1

|

=

|

2

n

+

1

|

=

2

n

+

1

<

ε

,

(4)

when

n

is chosen so big that

n

>

2

ε

-

1

.

(5)

The condition (5) is obtained from (4) by solving this

inequality

for

n

.  In this case, we have

N

=

2

ε

-

1

.

Example 2.

The so-called

decimal expansions

, i.e. endless

decimal numbers

, such as

3.14159265

⁢

…

=

π

,

0.636363

⁢

…

,

0.99999

⁢

…

,

(6)

are, as a matter of fact, limits of certain real number sequences.  E.g. the last of these is related to the sequence

0.9

,

0.99

,

0.999

,

…

(7)

which may be also written as

1

-

1

10

,

1

-

1

10

2

,

1

-

1

10

3

,

…

The limit of (7) is 1.  Actually, if

ε

>

0

,  the distance between 1 and the

n

th

member of (7) is

|

1

-

(

1

-

1

10

n

)

|

=

1

10

n

<

ε

,

when

10

n

>

1

ε

,  i.e. when

n

>

-

log

10

⁡

ε

=

N

.

The endless decimal notations (6) and others are, in fact, limit notations — no finite amount of decimals in them suffices to give their exact values.

Remark 2.

In both of the above examples, no of the sequence members was equal to the limit, but it does not need always to be so; thus for example

lim

n

→

∞

⁡

1

+

(

-

1

)

n

2

⁢

n

=

0

and every other member of the sequence in question is 0.

Infinite limits of real number sequences

There are sequences that have no limit at all, for example

1

,

-

1

,

1

,

-

1

,

1

,

-

1

,

…

.  Some real number sequences (1) have the property, that the member

a

n

may exeed every beforehand given real number

M

if one takes

n

greater than some value

N

(which depends on

M

):

a

n

>

M

when

n

>

N

.

Then we write

lim

n

→

∞

⁡

a

n

=

∞

.

Similarly, the sequence (1) may be such that for each

positive

M

there is

N

such that

a

n

<

-

M

when

n

>

N

,

and then we write

lim

n

→

∞

⁡

a

n

=

-

∞

.

E.g.

lim

n

→

∞

⁡

n

2

=

∞

,

lim

n

→

∞

⁡

(

1

-

n

)

=

-

∞

.

Title

limit of real number sequence

Canonical name

LimitOfRealNumberSequence

Date of creation

2015-01-30 17:53:27

Last modified on

2015-01-30 17:53:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 40A05

Synonym

limit of sequence of real numbers

Related topic

GeometricSequence

Related topic

BriggsianLogarithms

Related topic

InfiniteProductOfSums1a_i

Defines

limit

Generated on Fri Feb 9 21:14:57 2018 by

LaTeXML