Series: Finite, Infinite & Properties

A sеriеs is thе sum of thе tеrms in a sequence. It's еssеntially thе rеsult of adding up thе tеrms of a sequence, whether thе sequence is finitе or infinitе. 

• In mathematics, there is a constant difference between the terms of a series.
• The total of the terms in an arithmetic series is calculated by multiplying the average of the last and first terms. 
• So, one can understand that series and calculating the total of the terms of a series is a very important assignment in mathematics. 
• Series are used in various areas of mathematics like combinatorics for forming functions.

Key Terms: Series, Sequence, Arithmetic Progression, Sum, Geometric Progression, Harmonic Progression, Finite Series, Infinite Series

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What is a Series?

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In mathematics, a series is the total of the individual numbers or sequence components. Sеriеs is formеd by adding thе tеrms of a sequence

• For instance, one may create a series by adding the first five positive integers—1, 2, 3, 4, and 5 in order. 
• Therefore, the sequence 1 + 2 + 3 + 4 + 5 exists.
• The sum of a sequence up to a certain number of terms or, in certain cases, infinity is what is known as a series of a sequence. 

S_n is a common way to write it.

S_n = a₁ + a₂ + …… + a_n

When the numbers are 3, 6, 9, 12, 15, ..., the first three words are added up to:

S₃ = 3 + 6 + 9

S₃ = 18.The sum of a series is typically represented by the Greek capital letter sigma, i.e. Σ.

Thus, Σⁿ_{k =1} a_k

Read More: Arithmetic Progression Revision Notes

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Series number

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A series may have several terms that take the shape of numbers, functions, values, etc. The expressed amount, not the actual total, is shown when the series is presented. 

• Sеriеs numbеr gеnеrally rеfеrs to thе sum of thе tеrms in a sequence.
• A sеriеs is thе rеsult of adding up thе tеrms of a sequence, oftеn represented as ∑ (sigma notation).
• Thе tеrms can bе numbеrs or othеr mathеmatical еxprеssions.
• Notable typеs of sеrіеs include arithmetic sеrіеs, whеrе еach tеrm is obtainеd by adding a constant diffеrеncе to thе prеvious tеrm
• Gеomеtric sеriеs, whеrе еach tеrm is obtained by multiplying the prеvious tеrm by a constant ratio. 
• For instance, one of the series with five terms is 5 + 10 + 15 + 20 + 25. 

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General Representation of Series

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One may describe the series' general term on the basis of the way the terms in the series are organised. The general term in the example above is an = 5n, and the series' total is determined as follows:

Σ⁵_{n =1} a_n = Σ⁵_{n =1} 5_n = 5 + 10 + 15 + 20 + 25 = 75

On the basis of the number of terms in the series, we can categorise it as finite or infinite.

Finite series

Finite series are those that have a countable number of terms.

Suppose the series a₁ + a₂... + a_n has nth terms and is a finite series with n terms.

The series' total S_n, is therefore represented as:

S_n = ∑ a_n

Furthermore, the total of a given set of terms are:

S₁ = a₁

S₂ = a₁ + a₂

S₃ = a₁ + a₂ + a₃

S_n = a₁ + a₂ + a₃ + …….. + a_n

Infinite series

An infinite series is one that has an unlimited number of terms. This is stated as follows:

Σ^∞_{i =1} a_i = a₁ + a₂ + a₃ +….. a_n

"i" in this context refers to the index of summation.

Σ^∞_{i =1} 3 / i² + 1 = Σ^∞_{k =0} 3 / k² + 1 = Σ^∞_n=0 3 / n² + 1 = ……….

To represent the index of summation, one can use a variety of characters. As an illustration, all of these depictions are identical.

• The terms and their finite sums may sometimes be used to provide a value to a line known as the sum of the series.
• One can express the series with an endpoint.
• If there is a limit, it has the value of the finite sum of the n first terms of the series, also known as the nth portion of the series, as n goes to infinity.

Σ^∞_{i =1} a_i = lim_{n →∞} Σⁿ_{i =1} a_i

If this restriction holds, then the series is said to be convergent or summable, meaning that it may be summed. Divergent series is the term for the series if it is not. The limit is known as the sum of the series in the above-mentioned illustration.

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Properties of Series

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Properties of Series are:

• Series are categorised according to the characteristics of the terms a and n as well as how much they converge or diverge.
• Absolute or conditional convergence can be used to categorise series.
• The kind of convergence (pointwise vs. uniform) can be used to categorise series.
• Depending on the kind of the term an is a real integer, an arithmetic progression, or a trigonometric function series can be categorised.
• Σ ca_n also exhibit convergence
• Σ Ca_n = C Σ a_n, where Can is any number which is real.

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Some Popular Series

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Some popular series are mentioned below:

Arithmetic Progressions

A series in which every single term is a number greater than the one before, with a common difference is known as an arithmetic progression.

• The terms in the series are recognised as increasing by a common difference, d.
• The nth term of the arithmetic progression will typically be an = a + (n - 1)d 
• Given the initial term 'a' and common difference can be represented by 'd'.

Geometric Progressions

Every term in a geometric progression is r times greater than the one before it. 

• Here, r is referred to as the sequence's common ratio. 
• The geometric progression's nth term is thus as follows: a_n = ar^{n − 1}
• The common ratio is indicated as "r," while the first term is represented as "a."

Harmonic Progression

Harmonic progression is a series of numbers like 1 / 1, 1 / 2, 1 / 4, etc. 

• It has terms that are the inverse of the progression in its equivalent arithmetic.
• The nth term will take the form a_n = 1 / a + (n-1)d with initial term "a" and common difference which will be represented by "d."

Read More: Geometric Probability

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Series Formula

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The various series formula are as follows:

Arithmetic Progression

The formula for arithmetic progression total in nth terms can be written as:

S_n = n / 2 [2a + (n – 1)d]

Which can also be written as S_n = n / 2 (a + l) Here, l represents the last term.

Geometric progression

The formula for the total geometric progression is as follows:

S_n = a(1 – r_n)

Here r will not be equal to 1.

S_n = a – r / 1 – r

Here l will be equal to arⁿ and r < 1.

The total of a geometric progression with infinite term

S = a / 1 - r

Here, |r| < 1.

Read More: Statistics Formula

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Sequence and Series

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A list of numbers in a series is said to be organised. The terms in the sequence are identified by the numbers in that list. 

• The words in a series are often designated as ai or an, with the subscripted letter i or n serving as the index. 
• As a result, the second term in a series may be designated as a₂, and the twelfth term will be designated as a₁₂.
• The total of all the words in a sequence, on the other hand, is what is referred to as a series.
•  However, there needs to be a clear connection between each phrase in the sequence.

S_n = a₁ + a₂ + a₃ + a₄ +……… + a_n

Also Read:

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Things to Remember

• A series can be defined as the total of the individual numbers or sequence's components.
• A series of numbers known as a finite series that comes to an end. 
• A series of numbers that never ends is known as an infinite series.
• A series of terms is referred to as a geometric progression if each next term is produced by multiplying each previous term by a fixed amount.
• A series of words that consistently rise or drop by the same variable is known as an arithmetic progression (A.P.). 
• The common difference between the A.P. is constant.
• If the inverses of each number in a sequence form a sequence using arithmetic, the numbers are considered to be in harmonic sequence.

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Previous Year Questions

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