In mathematics, we can describe a series as adding infinitely many numbers or quantities to a given starting number or amount. We use series in many areas of mathematics, even for studying finite structures, for example, combinatorics for forming functions. The knowledge of the series is a significant part of calculus and its generalization as well as mathematical analysis. Apart from these applications in mathematics, infinite series are also extensively used in different quantitative disciplines such as statistics, physics, computer science, finance, etc.
Let’s have a look at the mathematical definition of the series given below.
Series Meaning
We can define series in maths based on the concept of sequences. As sequence and series are related concepts. Suppose a_{1}, a_{2}, a_{3}, …, a_{n} is a sequence such that the expression a_{1} + a_{2} + a_{3} +,…+ a_{n} is called the series associated with the given sequence. The series is finite or infinite, according to whether the given sequence is finite or infinite. Series are often represented in compact form, called sigma notation, using the Greek letter sigma, ∑ to indicate the summation involved.
Thus, the series a_{1} + a_{2} + a_{3} + … + a_{n} is abbreviated as [latex]\sum_{k=1}^{n}a_{k}[/latex].
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Series Number
A series may contain a number of terms in the form of numerical, functions, quantities, etc. When the series is given, it indicates the symbolised sum, not the sum itself. For example, 2 + 4 + 6 + 8 + 10 + 12 is a series with six terms. To find the sum of these numbers, we use the phrase “sum of a series”, which means the number that results from adding the terms of the series, the sum of the series is 42.
General Representation of a Series
Based on the pattern of terms in the series, we can define the general term of that series. In the above example, the general term is an = 2n and the sum of this series is given by:
[latex]\sum_{n=1}^{6}a_{n} = \sum_{n=1}^{6} 2n = 2 + 4 + 6 + 8 + 10 + 12 = 42[/latex]However, we can classify the series as finite and infinite based on the number of terms in it. These are explained below along with the formula, examples and properties.
Finite Series
A series with a countable number of terms is called a finite series.
If a_{1} + a_{2} + a_{3} + … + a_{n} is a series with n terms and is a finite series containing n terms.
Thus, S_{n} is the sum of the series and is denoted as:
S_{n} = ∑ a_{n}
Also, we can define the sum of a specific number of terms. These are expressed as:
S_{1} = a_{1}
S_{2} = a_{1} + a_{2}
S_{3} = a_{1} + a_{2} + a_{3}
S_{4} = a_{1} + a_{2} + a_{3} + a_{4}
….
S_{n} = a_{1} + a_{2} + a_{3} + … + a_{n}
Infinite Series
A series with an infinite number of terms is called an infinite series. This is expressed as:
[latex]\sum_{i=1}^{\infty }a_{i} =a_{1}+a_{2}+a_{3}+…+a_{n}+…[/latex]Here, “i” is called the index of summation.
We can use different letters to denote the index of summation. For example,
[latex]\sum_{i=1}^{\infty }\frac{3}{i^{2}+1}=\sum_{k=0}^{\infty }\frac{3}{k^{2}+1}= \sum_{n=0}^{\infty }\frac{3}{n^{2}+1}=…[/latex]All these representations are the same.
We cannot effectively carry the infinite string of additions mentioned by a series. Also, we can represent the series with a limit so that the set of terms and their finite sums are sometimes possible to assign a value to a string, called the sum of the series. If the limit exists, then its value is the limit as n tends to infinity the finite sums of the n first terms of the series, called the nth partial sums of the series.
[latex]\sum_{i=1}^{\infty } a_{i}=\lim_{n\rightarrow \infty }\sum_{i=1}^{n} a_{i}[/latex]A series is called convergent or summable if this limit exists, which means the sequence is summable. Otherwise, the series is called divergent series. In the above representation, the limit is called the sum of the series.
Properties of Series
Some of the properties of series are listed below:
If ∑a_{n} and ∑b_{n} are both convergent series, then
- ∑ ca_{n} is also convergent
- ∑ Ca_{n} = c ∑a_{n} such that c is any real number
- [latex]\sum_{n=k}^{\infty } a_{n}\pm \sum_{n=k}^{\infty } b_{n}[/latex] is also convergent
- [latex]\sum_{n=k}^{\infty } a_{n}\pm \sum_{n=k}^{\infty } b_{n} = \sum_{n=k}^{\infty }(a_{n}\pm b_{n})[/latex]
- [latex]\sum_{n=k}^{\infty } a_{n}\times \sum_{n=k}^{\infty } b_{n} \neq \sum_{n=k}^{\infty }(a_{n}\times b_{n})[/latex]
Solved Examples
Example 1:
If a_{n} = n(n + 2), then find the ∑a_{n} for 1 ≤ n ≤ 7.
Solution:
Given,
a_{n} = n(n + 2)
Substituting n = 1, 2, 3, …, 7
3, 8, 15, 24, 35, 48, 63
The series is: 3 + 8 + 15 + 24 + 35 + 48 + 63
∑a_{n} = 3 + 8 + 15 + 24 + 35 + 48 + 63 = 196
Example 2:
Find the sum of the following series:
[latex]1 + \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+…[/latex]Solution:
Given,
[latex]1 + \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+…[/latex]This is a geometric series with the general term a_{n} = 1/2^{n}.
Therefore, [latex]1 + \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+…=\sum_{n=0}^{\infty }\frac{1}{2^{n}}=2[/latex]
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