## JEE advanced important topics : Difference between a sequence and a series JEE advanced important topics : Difference between a sequence and a series

The terms series and sequence are used indistinguishably in common and informal practice. Nevertheless, they are very different concepts, especially when one takes into account their mathematical and scientific definitions. When considering this point of view, new factors appear, but first lest take a look at the basic definitions for each word:

SEQUENCE: a collection of numbers, or objects, that follow a definite pattern.

SERIES: addition of the members (terms) of a sequence.

So you see, to have a series you first need to have a sequence, they need each other. Then we can say that a list (it can be of anything you can imagine but lets say numbers) written in a definite order is called a sequence. When you add all the terms of an infinite sequence, it is called an infinite series. Therefore sequence is an ordered list of numbers and series is the sum of a list of numbers.

For example, lets study this sequence:

12, 14, 16, 18, 20 …

Then, if we add them up:

12+14+16+18+20+ …

what we now have is a series.

So whenever the word sequence is mentioned, you just have to think about a list of terms or numbers. In this case, the order of the numbers in the list is of particular importance. It has to follow a logic, a set of rules. For example, <1, 2, 3, 4> is a sequence of natural numbers from 1 to 4 in ascendant order. The list <4, 3, 2, 1> is another type of sequence, with the same natural numbers, but in descendant order. With this in mind, I am sure you can imagine a set of more complex and interesting sequences, like <2, 4, 8, 16, 32>, but it is very important to remember they must have a pattern. If they do not, they can not be considered a sequence.

This is specially important because, if you know the specific pattern a sequence follows, you can then predict what value, or number, is going to have in any of its terms. If you consider this sequence:

1 , 1/2, 1 /3, 1 /4, 1 /5 …

And someone tells you: what is the next element in the list? You can easily deduct that it has to be 1/6. Following the same pattern, you can find out that the value for the tenth term in the sequence is 1/10, and so on. This demonstrates that sequences have “behaviors”. In the last example the sequence goes from 1 to 1/5, so the behavior of the sequence gets closer and closer to zero. Since there is no negative value, or any number under zero in this sequences, then we assume the inferior limit, or end of the sequence, is zero.

On the contrary, a series consist simply in grouping or adding a list of numbers (for instance, 5 + 6 + 7 + 8 + 9). In a series, the order in which each element appears is not always important. This is because some series can have terms with no particular order or pattern, but they are still added together. This conform an absolutely convergent series. Nevertheles, they are also some series that change the order within the sum, changing the pattern.

Using the same example (the 1 to 1/5 sequence), if we associate it with a series, we could immediately say 1 + 1 /2 + 1/3 + 1 /4 + 1/5 and so on. If the answer or result of the series is too high, you write the infinite symbol, or, more appropriately, you can classify it as divergent.

### Mathematical definition of Sequence

In mathematics, an ordered set of numbers are said to be in a sequence if it has a definite value. The members of the sequence are called term or element. Every term in a sequence is related to the preceding and succeeding term. In general, sequences have a hidden rules, which helps you find out the value of the next term. The nth term is the function of integer n (positive), regarded as the general term of the sequence. A sequence can be finite or infinite.

Finite Sequence: A finite sequence is one that stops at the end of the list of numbers <1, 2, 3, 4, 5, 6……n>, is represented by: Infinite Sequence: An infinite sequence refers to a sequence which is unending, <1, 2, 3, 4, 5, 6…n…>, is represented by: Mathematical definition of Series

The addition of the terms of a sequence is known as series. Like sequence, series can also be finite or infinite, where a finite series is one that has a finite number of terms written as 1+2+3+4+5+6…n. Unlike infinite series, where the number of elements are not finite or which are unending, written as 1+2+3+4+5+6+…n+…

If 1+ 2+ 3+ 4+ 5+ 6 +…n= Sn, then Sn is considered as the sum to n elements of the series. The sum of terms is often represented by Greek letter sigma (Ʃ). Hence, Key differences between sequence and series

The difference between sequence and series can be listed as:

*The sequence is a collection of numbers that follow a pattern. When the elements of the sequence are added together, they are known as a series.

*Order matters in a sequence. Hence, <1, 2, 3> is different that <3, 2, 1>. On the other hand, in a series order may or may not matter.

Other mathematical examples

Arithmetic Progression (A.P.) and Geometric Progression (G.P.) are also sequences, not series. Arithmetic Progression is a sequence in which the pattern is a common difference between the consecutive terms such as 2, 4, 6, 8 and so on. On the contrary, in a geometric progression, each element of the sequence is the common multiple of the preceding term such as 3, 9, 27, 81 and so on. Similarly, Fibonacci Sequence is also one of the popular infinite sequence, in which each term is obtained by adding up the two preceding terms, like 1, 1, 3, 5, 8, 13, 21 and so on.