Power Series Examples

Power Series Calculator 

Know the power series representation of any function with this tool. The power series calculator has only four input fields and it provides the steps involved in finding each value of the series. 

How to use this tool?

The steps to find the power series from this calculator are:

  1. Enter the function.
  2. Select the variable.
  3. Input the enter point and the order.
  4. Click Calculate.

What is the power series? - Understanding its concept

Imagine you're trying to represent a complicated curve or shape using just a bunch of simple circles with different sizes and positions. It might sound impossible at first, but with enough circles and some tweaking, you can get pretty close to the original shape.

In mathematics, a power series is a bit like this idea, but instead of using circles, we're using polynomials (like x, x2, x3, etc.). Essentially, a power series is a way to represent functions as an infinite sum of these polynomials.

Here's the "recipe" for a power series:

f(x) = a0 + a1x +  a2x2 + a3x3 

Here: 

  • f(x) is the function to be represented.
  • The a0,a1,a2,… are the coefficients, which tell how much of each polynomial to use. 
  • x is the variable, like in a usual function.

By picking the right coefficients and adding up enough terms, we can make the polynomial sum resemble all sorts of different functions. The idea is that, with enough terms, we can get a good approximation of many functions.

How to expand the power series?

To expand a function into a power series means to express the function as a sum of powers of x. Here's how you can do it:

  • Find the value of the function at 0: This will be the constant term in our series.
  • Differentiate the function: This means finding a new function that describes the rate at which our original function changes.
  • Evaluate this new function at 0: This gives us the coefficient for the x term.
  • Keep differentiating: Each time you find a new rate of change, evaluate it at 0. This gives the next coefficient.
  • Assemble the terms together: Using the values found, start building the series.

Example:

Let's look at the function 

f(x) = ex :

  1. At x = 0, ex = e0 =1.
  2. Differentiate ex, and you still get ex . At x=0, this is again 1.
  3. Differentiate again. It's still ex. At x=0, it remains 1.

On continuing, you always get the same function and value. So, the series becomes:

ex  ≈ 1 + x + x2/2 + x3/6 +…

Where terms like x2/2 and x3/6 come from dividing by the number of times we've differentiated (2 times, 3 times, etc.) multiplied together.

In essence, we're building an "infinite polynomial" that mimics the original function, using its values and rates of change at 0.

Daily life applications:

Power series are incredibly versatile and play a significant role in many areas of mathematics and its applications. Here are some of the areas where power series are commonly used:

Solving Differential Equations: Many problems in physics and engineering lead to differential equations. For some of these equations, direct solutions are hard to find, but they can be approximated using power series.

Approximation: Power series can be used to approximate complex functions, especially when we are interested in values close to a specific point. This is useful in fields like numerical analysis.

Complex Analysis: In complex analysis, power series are used to represent functions of complex variables. They are crucial for understanding the properties and behaviors of complex functions.

Physics: Power series can be used to solve or approximate problems in quantum mechanics, electromagnetism, and other areas.

Engineering: In control theory and signal processing, power series are often used to analyze or design systems

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