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Difference Between Series and Sequence

Series vs Sequence

The terms “series” and “sequence” are often used interchangeably in common and non-formal practice. However, these terms are very distinct from each other with respect to mathematical and scientific viewpoints.

Foremost, when one talks about a sequence, it simply means a list or file of numbers or terms. So the order of the numbers in the list is of particular importance. It must be logical. For example, 6, 7, 8, 9, 10 is a sequence of numbers 6 to 10 in ascending order. The sequence 10, 9, 8, 7, 6 is another file that is arranged in descending order. There are other more complicated sequences that resemble some kind of pattern like 7, 6, 9, 8, 11, 10.

Because there is pattern in a sequence, one can readily guess the nth term. For example, in the sequence 1, 1/2, 1/3, 1/4, 1/5 and so on, if you are asked what the sixth 1/n term is, you can say that it is expected to be 1/6. The same pattern continues if you are asked for the one millionth nth term, it will be 1/1,000,000. This also shows that sequences have behaviors. In the above example of the sequence 1 to 1/5, the behavior of the sequence is moving closer to the zero value. However, as there will be no negative value or any number less than zero in the sequence, the limit or end of the sequence, no matter how long it will become, is assumed to be zero.

By contrast, a series is just adding up or summing a group of numbers (i.e., 6 + 7 + 8 + 9 + 10). Thus, a series has a sequence bearing terms (variables or constants) that were added. In a series, the order of appearance of each term is also important but not at all times as opposed to a sequence. This is because a few series can have terms without a particular order or pattern but will still add up together. These are termed as an absolutely convergent series. However, there are also some series that result in a change in the sum given a different type of order in the terms.

Using the same example (sequence 1 to 1/5), if you are to associate the sequence into a series, you can immediately write it as 1 + 1/2 + 1/3 + 1/4 + 1/5 and so on, and so forth. The answer or sum of the series is said to be very high. So it is described as infinite or, more appropriately, as divergent.

In summary, the two terms “series” and “sequence” are understandably causing much confusion to many. Nonetheless, it must be understood that:

1.The sum of the terms in the sequence is not a concern.
2.The sum of the terms in a series is of utmost concern.
3.The order or pattern of terms in a sequence is always important.
4.The order or pattern of terms in a series is sometimes important.
5.A sequence is a listing of numbers or terms while a series is the summation of the terms.

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