Converging and Diverging Sequences

Converging and diverging sequences are two different types of sequences in mathematics. A sequence is a list of numbers that follow a specific pattern or rule.

Converging Sequences:

A converging sequence is one in which the terms of the sequence approach a finite limit as the sequence progresses. In other words, the sequence gets closer and closer to a single number as the sequence continues.

For example, the sequence {1, 1/2, 1/3, 1/4, 1/5, ...} is a converging sequence because the terms of the sequence get smaller and smaller and approach zero as the sequence progresses. The limit of this sequence is zero.

Diverging Sequences:

A diverging sequence is one in which the terms of the sequence do not approach a finite limit as the sequence progresses. In other words, the sequence does not get closer and closer to a single number as the sequence continues.

For example, the sequence {1, 2, 3, 4, 5, ...} is a diverging sequence because the terms of the sequence continue to grow larger and larger without approaching a finite limit.

Series as Sum of Sequence:

A series is the sum of the terms of a sequence. For example, the series {1 + 1/2 + 1/3 + 1/4 + 1/5 + ...} is the sum of the terms of the converging sequence {1, 1/2, 1/3, 1/4, 1/5, ...}. This series is known as the harmonic series and diverges because the terms of the sequence do not approach a finite limit.

On the other hand, the series {1 - 1/2 + 1/3 - 1/4 + 1/5 - ...} is the sum of the terms of the alternating converging sequence {1, -1/2, 1/3, -1/4, 1/5, ...}. This series is known as the alternating harmonic series and converges to ln(2).

In conclusion, understanding the difference between converging and diverging sequences is important when dealing with series. A converging sequence will result in a converging series, while a diverging sequence will result in a diverging series.