Limit of Sequence

Sequence is a range of numbers / functions that have rules or ways of aligning them.

Example:
1, 3, 5, 7, ...
2, 4, 8, 16, ....

Each number is called the term of the sequence.

The general form:
u₁, u₂, u₃, ..., u_n or (u_n)
u_n = tribe-n
n = the number of tribes

If the numbers of the sequences are summed up there is a form called a series, which is usually abbreviated as S_n.

Example:
1 + 3 + 5 + 7 + ....
2 + 4 + 8 + 16 + ....

The general form:

The series is the number of rows.

Both the sequence and the series are functions of n or n variant, n is a tribe number.

Example:
sequence : f(n) = S_n = 3n - 1
Then the sequence is 2, 5, 8, 11, ...

L said limit (u_n), when for small positive numbers ε (eposilon) are defined, can be found index n₁ such that for each n > n₁ value apply |L-u_n| < ∈, written limit _n→∞ u_n = L

The meaning and the quantity of the definition can understand the following example:
The sequence (u_n), u_n = 2n/(n + 1), L = 2, ∈ = 1/1000, can it really be determined n₁ such that for n > n₁, applicable |2 - (2n/(n+1))| < 1/1000?

limit n→∞ (2n/(n+1)) = 2 → |2 - (2n/(n+1))| < 1/1000 → |(2n + 2 - 2n)/(n+1)| < 1/1000
|2/(n+1)| < 1/1000 → n + 1 > 2000 → n > 1999

It turns out to be found n₁ = 2000

The sequence is called convergent when the tribes have a limit (n → ∞) which finite, in other respects called divergent.

Example:
(u_n) = 1/2, 3/3, 5/4, 7/5, 9/6, ... , (2n - 1)/(n + 1).

The tribes of the ranks are increase, but limited.

limit _n→∞ u_n = limit _n→∞((2n - 1)/(n + 1)) = 2 → means (u_n), konvergen.