Sequences And Series: From Theory To Application.

This is an extensive course in sequences and series. You will explore how these fundamental mathematical concepts are used to model real-world phenomena. From the growth of populations to the spread of information, discover how sequences and series can be used to understand and predict patterns in diverse fields. You'll learn to identify, analyze, prove convergence of sequences and series, including infinite geometric series and solve problems involving arithmetic and geometric sequences and series, develop your problem-solving skills, and gain a deeper appreciation for the power of mathematics in everyday life. It is an essential building block for calculus and advanced mathematics. Through interactive lectures and problem-solving sessions, you will develop the ability to construct mathematical arguments and apply the epsilon-delta definition of convergence.As you delve deep into this course, you will learn the world of Fibonacci numbers and the Golden Ratio, examining their presence in nature, art, and mathematics. Students will learn about the history and properties of the Fibonacci sequence, explore its connections to the Golden Ratio, and discover its applications in various fields.       Let us, explicitly, define the constituent of this comprehensive course. It comprises in-depth knowledge of the course: use sequence notation to write terms of a sequence, write a rule for the nth term of a sequence, sum the terms of a sequence to obtain a series and use summation notation, learn about finite and infinite sequences and series, respectively ; use table of values and graphs to illustrate pattern of sequence , identify arithmetic sequences, write a rules for the nth term of arithmetic sequences, solve for a required term when two other terms are given, find partial sum and sums of finite arithmetic sequences using arithmetic progression formula, use sigma notation method to find the sum of arithmetic sequences and series ,use table of values and graphs to illustrate the concept of finite arithmetic sequence, model with arithmetic sequences and series, identify geometric sequences, write rules for geometric sequences, find partial sum and sums of finite geometric sequences using summation notation and to find sums using geometric progression formula, use table of values and graphs to illustrate the relationship between the terms of a given geometric sequence , Solve for the partial sum of infinite geometric series and sum of infinite geometric series by using formula of sum to infinite of geometric progression and use sigma notation to find the infinite sum of geometric sequences and series. Finally, there is a lecture on how to split infinite geometric series into two portions and solve for the sum to infinite of individual portions after which the two sums obtained are added together to get the required sum to infinite of the given special type of geometric series. This course offers a unique blend of mathematical exploration and real-world examples, making it both intellectually stimulating and visually engaging.