What are the practical applications of Sequences and Series?

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SUMMARY

Sequences and Series have significant practical applications across various fields, particularly in mathematics and engineering. Infinite series, such as the representation of repeating decimals and π, are foundational in calculus. Engineers frequently utilize infinite series to solve linear differential equations with variable coefficients. Additionally, power series, including Taylor and Maclaurin series, are crucial for approximating functions, while the Arctan series is notably used in sewing applications.

PREREQUISITES
  • Understanding of calculus concepts, specifically power series.
  • Familiarity with infinite series and their convergence properties.
  • Basic knowledge of differential equations and their solutions.
  • Introduction to complex algebra, particularly Laurent series.
NEXT STEPS
  • Research the applications of Taylor and Maclaurin series in function approximation.
  • Explore the role of Fourier Series in signal processing and data analysis.
  • Study the use of infinite series in solving linear differential equations.
  • Investigate the Residue Theorem and its applications in complex analysis.
USEFUL FOR

Mathematicians, engineers, students studying calculus, and anyone interested in the practical applications of sequences and series in real-world scenarios.

chound
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Does anyone know what are the uses of Sequences and Series in real life:confused:
 
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Have you come up with any examples yet?
 
What do you mean by "real life"? I use them all the time but then, I'm a mathematician so my "real life" might be different from yours! In other words, what we consider "uses" might be very different. I consider the fact that infinite decimal numbers, like the repeating decimal 0.33333... or \pi are defined in terms of infinite series to be an important use. An engineer might think of the fact that most linear differential equations with variable coefficients have to be solved using infinites series to be the most important use.

I think anyone who has taken enough calculus to work with power series (Taylor's or MacLaurin's series) would consider that to be an important application.
 
I need to prepare a term report i.e. a page on the applications of sequences and series
 
chound said:
Does anyone know what are the uses of Sequences and Series in real life:confused:
Actually, I heard that seamstresses are very fond of the Arctan series, because it resonates with the way they are stitching together clothes.

I'm not sure if it is really true, though.
 
Most of the functions can be written in Series like analytic ones in Power series,discontinuous ones in Fourier Series,As per according to Halls ,I found Power series method to very important.
And even in case of Complex Algebra ,Laurent series are helpful in finding real integrals making use of Residue theorem quite easily.
I believe they are very important tools of an Mathematician.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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