• Sequence
• Arithmetic
• Geometric
• nth
• Limit
• P.
• Monotone
• Bounded
• Convergent

Definition of the Limit of a Sequence

The limit of a sequence is the number which the terms of a sequence are approaching.

a₁= 1.

a₂= 0.5.

a₁₀₀₀= 0.001.

a_{1000 000} = 0.000001.

The limit is 0.

a₁= 0.5

a₂= 0.6666....

a₁₀₀₀= 0.999000999001

a_{1000 000} = 0.999999000001

The limit is 1.

a₁= 5

a₂= 7

a₁₀₀₀= 2,003

a_{1000 000} = 2,000,003

No particular number can represent the limit of this sequence, therefore, the limit is ∞.

Finite Limit of a Sequence

A squence, a_n, has a limit, L, if and only if for any positive number, ε, there is a term, a_k, from which all terms of a_n greater than a_k fulfill that |a_n−L| < ε.

The limit of the sequence a_n = 1/n is 0.

It can be determined from that term of the sequence that the distance from 0 is less than a positive number (ε).

From a₁₁, the distance to 0 is less than 0.1.

Determine from that term if the distance to 0 is less than 0.001.

From a₁₀₀₁, the distance to 0 is less than 0.001.

Infinite Limit of a Sequence

A sequence, a_n, has a limit of +∞ when for M > 0 there is a term, a_k, from which all the terms of a_n greater than a_k fulfill that a_n> M.

The limit of the sequence a_n = n² is +∞.

1, 4, 9, 16, 25, 36, 49, ...

If M = 10,000, its square root is 100, therefore, for a₁₀₁ it will exceed 10,000.

a₁₀₁= 101² = 10,201

A sequence, a_n, has a limit of −∞ when for N > 0 there is a term, a_k, from which all the terms of a_n greater than a_kfulfill that a_n < −N.

Verify that the limit of the sequence a_n = −n² is −∞.

−1, −4, −9, −16, −25, −36, −49, ...

If N = 10,000, its square root is 100, therefore, for a₁₀₁ it will exceed −10,000.

a₁₀₁= −101² = −10,201

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