What is the difference between a sequence and a series?

A sequence is an ordered list of numbers, while a series is the sum of those numbers.

How do I know if a series converges?

For infinite geometric series, if ( |r| < 1 ), it converges. Otherwise, check the limit of partial sums.

Can a harmonic series be summed to infinity?

No, the harmonic series diverges; its sum grows without bound.

Home

JEE Maths

Series

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Series: Types, Summation Formulas, Properties, Applications & Examples

Q: Can a harmonic series be summed to infinity?

No, the harmonic series diverges; its sum grows without bound.

1.0Introduction to Series

A series in mathematics is the sum of the terms of a sequence. While a sequence lists numbers in a specific order, a series focuses on their sum. In JEE Maths, series play a critical role in algebra, calculus, and problem-solving.

For example, the sum of the sequence 2, 4, 6, 8 is the series:
( 2 + 4 + 6 + 8 = 20 ).

Series can be finite (with a fixed number of terms) or infinite (continuing indefinitely). Mastery of series helps in evaluating complex expressions, solving calculus problems, and tackling various JEE questions efficiently.

2.0Types of Series

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic progression (AP), where each term increases by a constant difference.

General Form:

$(S_{n} = a + (a + d) + (a + 2 d) + ... + [a + (n - 1) d])$

Sum of n Terms:

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
or

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$

Example:
Sum of first 5 terms of 3, 7, 11, ...
( a = 3, d = 4, n = 5 )

$(S_{5} = \frac{5}{2} [2 \cdot 3 + (5 - 1) \cdot 4] = \frac{5}{2} [6 + 16] = \frac{5}{2} \cdot 22 = 55)$

Geometric Series

A geometric series is the sum of a geometric progression (GP), where each term is multiplied by a common ratio.

General Form:

$(S_{n} = a + a r + a r^{2} + ... + a r^{n - 1})$

Sum of n Terms $(f or (r \neq = 1))$ :

$S_{n} = a \frac{r ^{n} - 1}{r - 1}$

Sum to Infinity (for ( |r| < 1 )):

$S_{\infty} = \frac{a}{1 - r}$

Example:
( a = 2, r = 3, n = 4 )

$(S_{4} = 2 \frac{3 ^{4} - 1}{3 - 1} = 2 \frac{81 - 1}{2} = 2 \cdot 40 = 80)$

Harmonic Series

A harmonic series is formed by taking the sum of reciprocals of an arithmetic progression.

General Form:

$(S_{n} = \frac{1}{a} + \frac{1}{a + d} + \frac{1}{a + 2 d} + ... + \frac{1}{a + ( n - 1 ) d})$

The harmonic series diverges as $(n \to \infty)$ , i.e., its sum grows without bound.

Example:
The classic harmonic series: $(1 + \frac{1}{2} + \frac{1}{3} + ...)$

Infinite Series

A series with an infinite number of terms.
Examples:

$Convergent: \sum \frac{1}{2 ^{n}} = 1$

$Divergent: \sum \frac{1}{n} (Harmonic series)$

Special Series

i. Series of Natural Numbers:

$(1 + 2 + 3 + ... + n = \frac{n ( n + 1 )}{2})$

ii. Series of Squares:

$(1^{2} + 2^{2} + 3^{2} + ... + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6})$

iii. Series of Cubes:

$(1^{3} + 2^{3} + 3^{3} + ... + n^{3} = [\frac{n ( n + 1 )}{2}]^{2})$

iv. Telescoping Series:
A series where many terms cancel each other when expanded, simplifying the sum.

3.0Summation of Series

Summation Notation

The symbol $(\sum) (σ)$ represents the sum of a sequence.

General Notation:

$(\sum_{k = 1}^{n} a_{k} = a_{1} + a_{2} + \dots + a_{n})$

Example:

$(\sum_{k = 1}^{5} k = 1 + 2 + 3 + 4 + 5 = 15)$

Common Summation Formulas

$(\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2})$
$(\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6})$
$(\sum_{k = 1}^{n} k^{3} = [\frac{n ( n + 1 )}{2}]^{2})$
$(\sum_{k = 0}^{n - 1} a r^{k} = a \frac{r ^{n} - 1}{r - 1}) (GP)$

4.0Properties of Series

Linearity:

$(\sum (a_{k} + b_{k}) = \sum a_{k} + \sum b_{k})$

$(\sum c \cdot a_{k} = c \sum a_{k})$

Splitting:
A series can often be split into simpler parts for easier calculation.
Convergence/Divergence:
For infinite series, convergence determines if the sum approaches a finite value.
Telescoping:
A telescoping series collapses to just a few terms after cancellation.

5.0Sum of Series: Formulas and Methods

Knowing how to sum different types of series is essential for JEE.

Arithmetic Series:

$[S_{n} = \frac{n}{2} [2 a + (n - 1) d]]$

Geometric Series:

$[S_{n} = a \frac{1 - r ^{n}}{1 - r}]$

Sum of First ( n ) Natural Numbers:

$[1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}]$

Sum of Squares:

$[1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}]$

Sum of Cubes:

$[1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = [\frac{n ( n + 1 )}{2}]^{2}]$

Telescoping Series:

If $(S = (a_{1} - a_{2}) + (a_{2} - a_{3}) + \dots + (a_{n} - a n + 1))$ ,
Then $(S = a_{1} - a n + 1)$ .

6.0Solved Examples on Series

Example 1: Arithmetic Series

Q: Find the sum of the first 20 terms of the series 5, 8, 11, ...

A:
( a = 5, d = 3, n = 20 )

$(S_{20} = \frac{20}{2} [2 \cdot 5 + (20 - 1) \cdot 3] = 10 [10 + 57] = 10 \times 67 = 670)$

Example 2: Geometric Series

Q: Find the sum of first 6 terms of GP: 3, 6, 12, 24, ...

A:
( a = 3, r = 2, n = 6 )

$(S_{6} = 3 \frac{2 ^{6} - 1}{2 - 1} = 3 \times (64 - 1) = 3 \times 63 = 189)$

Example 3: Series of Squares

Q: Calculate $(1^{2} + 2^{2} + 3^{2} + ... + 1 0^{2})$ .

A:

$(S = \frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385)$

Example 4: Infinite Geometric Series

Q: Find the sum of the infinite series: $(6 + 2 + \frac{2}{3} + ...)$

A:

$(a = 6, r = \frac{1}{3})$

$(S_{\infty} = \frac{6}{1 - \frac{1}{3}} = \frac{6}{\frac{2}{3}} = 9)$

Example 5: Telescoping Series

Q: Evaluate $(\sum_{k = 1}^{10} \frac{1}{k ( k + 1 )})$ .

A: $(\frac{1}{k ( k + 1 )} = \frac{1}{k} - \frac{1}{k + 1})$
So,

$S = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + ... + (\frac{1}{10} - \frac{1}{11})$
All intermediate terms cancel, leaving:

$1 - \frac{1}{11} = \frac{10}{11}$

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Series: Types, Summation Formulas, Properties, Applications & Examples

1.0Introduction to Series

A series in mathematics is the sum of the terms of a sequence. While a sequence lists numbers in a specific order, a series focuses on their sum. In JEE Maths, series play a critical role in algebra, calculus, and problem-solving.

For example, the sum of the sequence 2, 4, 6, 8 is the series:
( 2 + 4 + 6 + 8 = 20 ).

Series can be finite (with a fixed number of terms) or infinite (continuing indefinitely). Mastery of series helps in evaluating complex expressions, solving calculus problems, and tackling various JEE questions efficiently.

2.0Types of Series

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic progression (AP), where each term increases by a constant difference.

General Form:

$(S_{n} = a + (a + d) + (a + 2 d) + ... + [a + (n - 1) d])$

Sum of n Terms:

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
or

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$

Example:
Sum of first 5 terms of 3, 7, 11, ...
( a = 3, d = 4, n = 5 )

$(S_{5} = \frac{5}{2} [2 \cdot 3 + (5 - 1) \cdot 4] = \frac{5}{2} [6 + 16] = \frac{5}{2} \cdot 22 = 55)$

Geometric Series

A geometric series is the sum of a geometric progression (GP), where each term is multiplied by a common ratio.

General Form:

$(S_{n} = a + a r + a r^{2} + ... + a r^{n - 1})$

Sum of n Terms $(f or (r \neq = 1))$ :

$S_{n} = a \frac{r ^{n} - 1}{r - 1}$

Sum to Infinity (for ( |r| < 1 )):

$S_{\infty} = \frac{a}{1 - r}$

Example:
( a = 2, r = 3, n = 4 )

$(S_{4} = 2 \frac{3 ^{4} - 1}{3 - 1} = 2 \frac{81 - 1}{2} = 2 \cdot 40 = 80)$

Harmonic Series

A harmonic series is formed by taking the sum of reciprocals of an arithmetic progression.

General Form:

$(S_{n} = \frac{1}{a} + \frac{1}{a + d} + \frac{1}{a + 2 d} + ... + \frac{1}{a + ( n - 1 ) d})$

The harmonic series diverges as $(n \to \infty)$ , i.e., its sum grows without bound.

Example:
The classic harmonic series: $(1 + \frac{1}{2} + \frac{1}{3} + ...)$

Infinite Series

A series with an infinite number of terms.
Examples:

$Convergent: \sum \frac{1}{2 ^{n}} = 1$

$Divergent: \sum \frac{1}{n} (Harmonic series)$

Special Series

i. Series of Natural Numbers:

$(1 + 2 + 3 + ... + n = \frac{n ( n + 1 )}{2})$

ii. Series of Squares:

$(1^{2} + 2^{2} + 3^{2} + ... + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6})$

iii. Series of Cubes:

$(1^{3} + 2^{3} + 3^{3} + ... + n^{3} = [\frac{n ( n + 1 )}{2}]^{2})$

iv. Telescoping Series:
A series where many terms cancel each other when expanded, simplifying the sum.

3.0Summation of Series

Summation Notation

The symbol $(\sum) (σ)$ represents the sum of a sequence.

General Notation:

$(\sum_{k = 1}^{n} a_{k} = a_{1} + a_{2} + \dots + a_{n})$

Example:

$(\sum_{k = 1}^{5} k = 1 + 2 + 3 + 4 + 5 = 15)$

Common Summation Formulas

$(\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2})$
$(\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6})$
$(\sum_{k = 1}^{n} k^{3} = [\frac{n ( n + 1 )}{2}]^{2})$
$(\sum_{k = 0}^{n - 1} a r^{k} = a \frac{r ^{n} - 1}{r - 1}) (GP)$

4.0Properties of Series

Linearity:

$(\sum (a_{k} + b_{k}) = \sum a_{k} + \sum b_{k})$

$(\sum c \cdot a_{k} = c \sum a_{k})$

Splitting:
A series can often be split into simpler parts for easier calculation.
Convergence/Divergence:
For infinite series, convergence determines if the sum approaches a finite value.
Telescoping:
A telescoping series collapses to just a few terms after cancellation.

5.0Sum of Series: Formulas and Methods

Knowing how to sum different types of series is essential for JEE.

Arithmetic Series:

$[S_{n} = \frac{n}{2} [2 a + (n - 1) d]]$

Geometric Series:

$[S_{n} = a \frac{1 - r ^{n}}{1 - r}]$

Sum of First ( n ) Natural Numbers:

$[1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}]$

Sum of Squares:

$[1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}]$

Sum of Cubes:

$[1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = [\frac{n ( n + 1 )}{2}]^{2}]$

Telescoping Series:

If $(S = (a_{1} - a_{2}) + (a_{2} - a_{3}) + \dots + (a_{n} - a n + 1))$ ,
Then $(S = a_{1} - a n + 1)$ .

6.0Solved Examples on Series

Example 1: Arithmetic Series

Q: Find the sum of the first 20 terms of the series 5, 8, 11, ...

A:
( a = 5, d = 3, n = 20 )

$(S_{20} = \frac{20}{2} [2 \cdot 5 + (20 - 1) \cdot 3] = 10 [10 + 57] = 10 \times 67 = 670)$

Example 2: Geometric Series

Q: Find the sum of first 6 terms of GP: 3, 6, 12, 24, ...

A:
( a = 3, r = 2, n = 6 )

$(S_{6} = 3 \frac{2 ^{6} - 1}{2 - 1} = 3 \times (64 - 1) = 3 \times 63 = 189)$

Example 3: Series of Squares

Q: Calculate $(1^{2} + 2^{2} + 3^{2} + ... + 1 0^{2})$ .

A:

$(S = \frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385)$

Example 4: Infinite Geometric Series

Q: Find the sum of the infinite series: $(6 + 2 + \frac{2}{3} + ...)$

A:

$(a = 6, r = \frac{1}{3})$

$(S_{\infty} = \frac{6}{1 - \frac{1}{3}} = \frac{6}{\frac{2}{3}} = 9)$

Example 5: Telescoping Series

Q: Evaluate $(\sum_{k = 1}^{10} \frac{1}{k ( k + 1 )})$ .

A: $(\frac{1}{k ( k + 1 )} = \frac{1}{k} - \frac{1}{k + 1})$
So,

$S = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + ... + (\frac{1}{10} - \frac{1}{11})$
All intermediate terms cancel, leaving:

$1 - \frac{1}{11} = \frac{10}{11}$

Frequently Asked Questions

What is the difference between a sequence and a series?

How do I know if a series converges?

Can a harmonic series be summed to infinity?

Frequently Asked Questions

What is the difference between a sequence and a series?

How do I know if a series converges?

Can a harmonic series be summed to infinity?

Join ALLEN!

Series: Types, Summation Formulas, Properties, Applications & Examples

1.0Introduction to Series

2.0Types of Series

Arithmetic Series

Geometric Series

Harmonic Series

Infinite Series

Special Series

3.0Summation of Series

Summation Notation

Common Summation Formulas

4.0Properties of Series

5.0Sum of Series: Formulas and Methods

6.0Solved Examples on Series

Example 1: Arithmetic Series

Example 2: Geometric Series

Example 3: Series of Squares

Example 4: Infinite Geometric Series

Example 5: Telescoping Series

Table of Contents

Join ALLEN!

Series: Types, Summation Formulas, Properties, Applications & Examples

1.0Introduction to Series

2.0Types of Series

Arithmetic Series

Geometric Series

Harmonic Series

Infinite Series

Special Series

3.0Summation of Series

Summation Notation

Common Summation Formulas

4.0Properties of Series

5.0Sum of Series: Formulas and Methods

6.0Solved Examples on Series

Example 1: Arithmetic Series

Example 2: Geometric Series

Example 3: Series of Squares

Example 4: Infinite Geometric Series

Example 5: Telescoping Series

Table of Contents