Advanced Algebra : Some Special Series of Algebra

Name: Advanced Algebra : Some Special Series of Algebra
Rating: 4.4 (7 reviews)

Special Series of Algebra-AGP,, Series related to natural nos, Method of difference, Exponential and Logarithmic

Created bySandeep Kumar Mathur

Last updated 7/2026

English

English [Auto],

What you'll learn

Students will learn complete knowledge of exponential series
Students will learn complete knowledge of Logarithmic series
Students will learn basic concepts, Formulae and Important results
You will understand solution of selected questions step by step

Course content

6 sections • 60 lectures • 5h 33m total length

Formulae of Arithmetic Progressions4:55
Example-14:06
Example-22:54
Determine the nth term a_n = 4n + 5 and verify the resulting sequence is an arithmetic progression with a common difference of 4.
Example-33:49
Identify the sequence 4, 9, 14, 19 as an arithmetic progression with a = 4 and d = 5. Set 124 = 4 + (n-1)5 to find n = 25.
Example-43:17
Example-51:55
Formulae of Geometric Progressions3:39
Recall the formulae for geometric progressions: first term a, common ratio r, nth term t_n = a r^{n-1}, and finite and infinite sums when |r|<1.
Example-14:40
The lecture analyzes a geometric progression with first term 1/4 and common ratio -2, derives t_n = a r^(n-1), and computes t9 = 64 and the general term as (1/4)(-2)^(n-1).
Example-23:48
Find the fourth term from the end of the geometric progression 3, 6, 12, 24, ..., 3072 using r = 1/2 and T_n = a r^(n-1), yielding 384.
Example-34:21
In this example, identify the ninth term of the geometric progression 2, 1/2, 1/4, ..., by setting t_n = a r^(n-1) equal to 1/128 and solving for n.
Example-42:53
Example-52:44
Prove that the infinite product 6^(1/2) × 6^(1/4) × 6^(1/8) × … equals 6 by summing the exponents in a geometric progression with ratio 1/2, showing convergence and equality.
Example-62:11
Compute the sum of seven terms of the GP 3, 6, 12, ... using a = 3 and common ratio r = 2, yielding 381.
Factorial Notations3:55
Review factorial notation, including n!, and the relation n! = n*(n-1)!, with 0! = 1 and undefined for negative numbers or fractions, and its use in exponential series.
Example-12:49
Solve for x in the equation x over six factorial equals one over four factorial plus one over five factorial by rewriting factorials and cross-multiplying to obtain x equals 36.
Example-22:30
Cancel factorials to simplify expressions: factorial 30 over factorial 28 equals 30*29=870. The expression (factorial 11 minus factorial 10) over factorial 9 simplifies to 100.
Example-32:25
Compare left-hand side and right-hand side factorial expressions to determine truth, showing why factorial five differs from the sum of factorial two and factorial three, and why the statements are false.
Example-43:22
Example-54:02
compare factorial expressions n!/(2!(n-2)!) and n!/(4!(n-4)!), simplify, cancel common factors, and solve n(n-1)=6 to find n=5.
Formulae of Logarithms4:02
Recall logarithm formulae and laws: log m base a + log n base a = log mn base a; log m base a = log m / log a.
Example-13:09
Example-21:30
Example-34:10
Example-44:34
Example-56:27

Basic Concepts18:31
Example - 111:32
Explore the arithmetico geometric progression in the series 1, 3x, 5x^2, 7x^3, ..., derive nth terms of the AP and GP, and determine the sum s_n.
Example - 212:03
Example - 35:35
Explore the sum to infinity of the series 1 + 2x + 3x^2 + 4x^3 + ... for modulus x < 1, showing it equals 1/(1-x)^2.
Example - 46:25
Find the sum to infinity of a series modeled as a geometric progression with ratio minus x, for |x|<1, yielding (1 - x)/(1 + x)^2.
Example - 510:01
Solve for the common ratio in the infinite series 3 + 5r + 7r^2 + ... with sum 44/9, using s_infinity = a/(1−r); find r = 1/4 after rejecting 17/11.
Example - 66:49
Apply the sum-to-infinity formula for a geometric progression with ratio 1/5 to obtain the infinite sum 15/8 and solve for d.
Example - 714:29
Compute the sum to n terms of the arithmetico-geometric progression with nth term (3n-1)x^(n-1), derive s_n, then deduce the infinite sum for |x|<1 as (2+x)/(1−x)^2.
Example - 87:41
Demonstrates finding the sum to infinity of 1^2 + 2^2 x + 3^2 x^2 + 4^2 x^3 + ... by using an arithmetico-geometric progression, yielding s_infinity = (1+x)/(1−x)^3.
Example - 98:18
Analyze the sum to infinity of the series 1^2 + 5^2 x + 9^2 x^2 + 13^2 x^3 for |x|<1. It forms an arithmetico geometric progression rather than a GP.

Basic Concepts6:05
Explore the fundamental formulas for sums of first n natural numbers, squares, and cubes, including sigma n, sigma n square, and sigma n cube, with practical examples.
Example - 15:16
Express the series of odd squares as (2k-1)^2, expand, and simplify to obtain S_n = n(4n^2-1)/3.
Example - 21:20
Example-36:30
Derive the sum of the series of consecutive triple products k(k+1)(k+2) up to n terms by finding the nth term and using sigma formulas to obtain s_n = n(n+1)(n+2)(n+3)/4.
Example-46:13
Example-53:56
Example-67:02

Basic Concepts1:25
Apply the method of difference to find t_n when the nth term is unknown, then use ap and gp formulae to compute t_n and s_n up to n terms.
Example-17:20
Apply the method of differences to find the nth term and the sum to n terms of the series 3, 15, 35, 63, using an arithmetic progression approach.
Example-25:56
Apply the method of differences to the series 1, 5, 12, 22, 35 to obtain tn and derive Sn = n^2(n+1)/2 for the first n terms.
Example-36:09
this lecture demonstrates the method of differences to find t_n for a series forming a geometric progression, yielding t_n = 3^{n-1} + 4 and s_n = (3^n - 1)/2 + 4n.

Basic Concepts11:24
Example-12:59
Compute the coefficient of x^10 in the expansion of e^{2x} using the series e^x = sum x^n/n!, showing the coefficient equals 2^{10}/10!.
Example-22:17
Example-31:41
Expand e^{2x+3} as e^{2x} e^{3}, apply the series for e^{2x}, and extract the x^2 term to obtain 2 e^3.
Example-41:58
Show that the coefficient of x^n in the series is e^a b^n / n!, obtained by treating the series as e^a e^{b x} and expanding.
Example-515:35
Learn to sum infinite series with factorial denominators and express results in e. Derive forms such as e minus 2, e minus 5/2, and e minus e^{-1}/2 for nth-term patterns.
Example-66:14
Derive the sum to infinity of a special series by identifying the nth term, manipulating factorials, and proving the total equals three e.
Example - 73:21
Analyze the series with the nth term n squared over n factorial, rewrite terms to form two copies of the e-series, and conclude the sum equals 2e.
Example-89:33
Determine the nth term of a series with numerator 1+2+...+n over n!, then sum to infinity to obtain 3e/2.
Quiz(Exponential Series)

Basic Concepts11:08
Logarithmic series
Example-13:16
From the series y = x - x^2/2 + x^3/3 - ..., for |x|<1, derive x = y + y^2/2! + y^3/3! + ..., via log(1+x) and e^y expansions.
Example - 25:49
The lecture shows that the left-hand series equals log base e of a minus log base e of b, via the expansion of log(1+x) and log(1−x) with x = (a−b)/a.
Example - 35:45
An example to understand the concepts....
Example - 45:18
An example to understand the concepts....
Quiz(Logarithmic Series)

Requirements

Factorial notations,sequence and series,laws of Logarithms and elementary algebra

Description

Unlock the Power of Special Series: Master Advanced Algebra

Are you ready to elevate your mathematical prowess? Dive deep into the captivating world of Advanced Algebra: Special Series and conquer complex mathematical concepts with confidence. This comprehensive course is meticulously designed for students aiming for competitive exams and those eager to explore the intricate beauty of advanced algebra.

What You'll Master:

Arithmetic Geometric Series: Learn to tackle series that blend arithmetic and geometric progressions, unraveling their unique patterns and summation techniques.
Series of n Natural Numbers: Master the art of summing powers of natural numbers, equipping yourself with essential formulas and problem-solving strategies.
Method of Difference: Discover this powerful technique for handling series where differences between consecutive terms reveal hidden patterns, enabling you to derive general terms and sums.
Exponential Series: Explore the fascinating world of exponential series, understanding their convergence, applications, and connections to exponential functions.
Logarithmic Series: Delve into the intricacies of logarithmic series, mastering their manipulation, and applications in various mathematical contexts.

Why Choose This Course?

Comprehensive Content: Each topic is thoroughly explained with numerous illustrative examples, ensuring a solid understanding of the underlying principles.
Interactive Learning: Reinforce your knowledge with engaging quizzes designed to test your comprehension and sharpen your problem-solving skills.
Competitive Edge: Gain a significant advantage in competitive exams by mastering these crucial advanced algebra topics.
Robust Platform: We are dedicated to building a strong learning environment for all of our students.
Dedicated Q&A Support: We understand that learning can be challenging. Our dedicated Q&A support ensures that all your queries are addressed promptly, fostering a supportive and interactive learning experience.
Boost Your Mathematical Confidence: This course is designed to significantly enhance your understanding of exponential and logarithmic functions, building your overall mathematical confidence.

Our Goal:

We are committed to providing a robust platform for students seeking to excel in competitive exams and deepen their understanding of mathematics. We believe this course has the potential to significantly enhance your understanding of exponential and logarithmic functions while boosting your overall mathematical confidence.

Ready to embark on this exciting learning adventure?

Don't miss this opportunity to master advanced algebra and unlock your full mathematical potential. Join us now!

Who this course is for:

class 11th students and those who are preparing for IIT JEE,NDA exams.

Advanced Algebra : Some Special Series of Algebra

What you'll learn

Explore related topics

Course content

Preliminaries25 lectures • 1hr 28min

Arithmetico Geometric Progressions10 lectures • 1hr 41min

Sum of n terms of special sequences related to natural numbers7 lectures • 36min

Method of Difference4 lectures • 21min

Exponential Series9 lectures • 55min

Logarithmic Series5 lectures • 31min

Requirements

Description

Who this course is for: