How Are Power and Taylor Series Used in Real-World Applications?

[jabers]

In the real world what are power or taylor series used for? Historically were they used for anything? Especially were they used for anything interesting?

 

[CRGreathouse]

In the real world what are power or taylor series used for?

Calculating things that are too complicated to calculate in other ways, or working mathematically with objects that are too complicated to analyze in other ways.

 

[chiro]

In the real world what are power or taylor series used for? Historically were they used for anything? Especially were they used for anything interesting?

They have quite a variety of uses. Let's look at a few examples:

1) The transcendental functions (ie sin, cos, tan, log, exp etc)

We know from taylor series that we can represent a function by the rgelationship to its derivatives and function value at a point.

Now we don't know how to calculate sin(x) or cos(x) but we know the derivatives of these functions and their values at x = 0.

Using a special case of taylor series (called a mclaurin series) we can find an expression for sin(x) when x <> 0 using knowledge about the differential at various degrees.

So all of the transcendental functions can be calculated to find the value to any desired approximation.

Also you should note that any function that has infinite terms has the potential to have infinite stationary points (turning points or points of inflection), so anything that is periodic over an infinite domain is basically a series. This brings me to part 2:

2) Fourier series:

Fourier series builds on the idea that we can take things from the time domain and put them into the frequency domain.

A lot of functions that a periodic over the reals have surprising simple series representations. Examples of this include the sawtooth function, the "clock" function, the signum function and so on.

All of the above functions can be represented by infinite series and we can get as good approximations as we want to these with series expressions.

3) Systems in math and nature:

The fact is that a lot of different systems do not have a closed form answer: they can be written in terms of infinite series.

One surprising kind of math that uses an infinite series is called the Riemann Zeta Function. It has connections everywhere including number theory and even physics. There is a one million dollar reward to prove that the non trivial zeroes have real part = 1/2.

If you look at many areas of science (including physics) you will see many examples of systems that have these so called series expansions.

I hope that gives some insight to what is out there with series

 

[michaelc187]

nothing can done without them, not even calculus.

 

[CellCoree]

Series are mathematical tools used to represent a function as an infinite sum of simpler functions. They are commonly used in various fields of science, including physics, engineering, and economics, to model and analyze complex systems.

In the real world, power series and Taylor series are commonly used in physics and engineering to approximate and solve differential equations, which are fundamental in understanding the behavior of physical systems. For example, power series are used to approximate the solutions to problems in electromagnetism and fluid dynamics, while Taylor series are used in the study of heat transfer and vibration analysis.

Historically, power and Taylor series were first introduced by mathematicians in the 17th century, but their applications in science and engineering were not fully realized until the 19th and 20th centuries. They have been used in various interesting applications, such as predicting planetary motion, analyzing the behavior of electrical circuits, and modeling the spread of diseases.

One particularly interesting application of power and Taylor series is in the study of fractals, which are complex geometric patterns found in nature. These series are used to represent and analyze the self-similar nature of fractals, allowing scientists to better understand and predict their behavior.

In conclusion, series, particularly power and Taylor series, have a wide range of applications in the real world, from solving differential equations to studying complex systems and phenomena. They have played a significant role in advancing our understanding of the natural world and continue to be a valuable tool in scientific research.