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Sequences and Series

A sequence is a collection of objects listed in a specific order.
That is a clear order of terms as first, second, third etc. can be found.

A sequence can also be viewed as a function whose domain is a set of consecutive integers which refer to the position of the terms in the sequence.
If the domain is not explicitly specified, it can be taken that the domain starts with 1.
The range of the function is the corresponding values or terms in the sequence.
Domain:  1, 2, 3, ..........n               Numbers referring to the positions
Range:  a1, a2, a3, ..........an            The terms forming the sequence.
Example:
In the sequence 3, 6, 9, 12, 15, 18  there are six terms.
a1 = 3    a2 = 6   a3 =  9   a4 = 12     a5 = 15     a6 = 18
Sequences like the above with a fixed or limited number of terms are called as finite series.
A sequence where the list of terms progress without stopping is called an infinite series.

Example:
2, -4, 8, -16, 32,..........

A series is an expression resulting by adding the terms of an sequence. 
A series can also be either finite or infinite depending upon the sequence whose terms are added to form the Series.

The finite series corresponding to sequence 3, 6, 9, 12, 15, 18  is 3 + 6 + 9 + 12 + 15 + 18 =  63.
An infinite series is in turn categorized as Convergent and Divergent series depending on the sum converges to a finite sum or not.

Related Calculators
Sequence Calculator Arithmetic Sequence Calculator
Fibonacci Sequence Calculator Geometric Sequence Calculator
 

Sequences and Series Examples

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A sequence is often described by a rule for the general term. The sequence can be expanded by substituting the position number in the rule.
Example:
Write the first four terms of the sequence represented by an = 3n - 5.
Substituting n = 1, 2, 3 and 4 in the above equation we get
First term           a1 = 3(1) - 5 = 3 - 5 = -2
Second term      a2 = 3(2) - 5 = 6 - 5 = 1
Third term          a3 = 3(3) - 5 = 9 - 5 = 4
Fourth term        a4 = 3(4) - 5 = 12 - 5 = 7

It is often required to find the rule for the sequence from the given list of first few terms.
Find the rule for the sequence 1, 3, 5, 7,........
The pattern in the first four terms of the sequence can be written as...
a1 = 1 = 2(1) - 1
a2 = 3 = 2(2) - 1
a3 = 5 = 2(3) - 1
a4 = 7 = 2(4) - 1
Each term in the sequence is got by multiplying its position number by 2 and then subtracting 1.
We can extend this pattern to the general term as an = 2n -1

Formulas for some special series
If a series contains a small number of terms we can find the sum by adding the terms. But a formula, if it could be found is useful in evaluating a series with a large number of terms. The commonly used formulas for some special series are given below:

1. Sum of n terms of a series with repeating terms
    a + a + a + ...........+ n terms = na

2. Sum of the first n positive integers
    1 + 2 + 3 + ..........+ n = $\frac{n(n+1)}{2}$

3. Sum of the squares of first n positive integers
    12 + 22 + 32 + ......... + n2 = $\frac{n(n+1)(2n+1)}{6}$

4. Sum of the cubes of first n positive integers
    13 + 23 + 33 +............+ n3 = $\frac{n^{2}(n+1)^{2}}{4}$


More topics in  Sequences and Series
Arithmetic Series Geometric Series
Harmonic Series Taylor Series
Fourier Series
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