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Power Series Calculator

Enter the values into this calculator to find the power series expansion of the function around the given point and up to order (n).

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Power Series Calculator:

This power series calculator allows you to expand a function into a power series with respect to a given variable. It lets you make calculations by:

  • Performing power series expansion for a function.
  • Specify the point (center) around which you want to expand.
  • Setting the order (n) of the expansion.

Using this calculator helps for analyzing and approximating the function values. 

Limitation: The calculator can handle a range of mathematical functions while this might not work for functions with discontinuities or infinite complexity.

What is Power Series?

"A power series is an infinite series where every single term is a constant that is multiplied by the variable (x) to an increasing non-negative power (n). It proceeds to represent a function within the interval of convergence"

The series behaves as a function along its convergence and divergence. Convergent values are determined based on the selected x-value. 

Power Series defines new functions and also used to show the common functions. Also, this term approximates the functions, solves differential equations, and evaluates integrals

Power Series Formula:

According to the definition, a power series (in one variable) is indicated as an infinite series of the form. A general equation is:

\(\ \sum_{n=0}^{\infty} a_n (x-a)^n = a^0 + a_1x + a_2 x^2 + a_3 x^3 + ...\)

Where:

  • \(\ \sum\) = Summation which means summing up numbers at infinity
  • \(\ a_n\) = coefficient of the nth term from real to complex
  • \(\ x\) = variable
  • \(\ n\) = exponent ranging from 0 to infinity

Power Series Convergence:

A power series centered at a converges for a value of x within a certain interval and the terms get smaller. Hence, their sum approaches a finite value. This convergence can be found by using the ratio test. The convergence of power series is also known as the radius of convergence. 

\(\ {\sum\limits ({x^n}}) = 1 + x + x2 + x3 + ...\)

\(\ \sum_{n=0}^{\infty} x^n\)

The power series converges at when the absolute value |x| < 1.

At this point, its value becomes \(\ \frac{1}{1 - x}\). We can express the function to power series as below. Also, our power series calculator takes a function and converts it into its equivalent power series representation.

\(\ f(x) = \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n} = 1 + x + x^2 + x^3 + \dots\)

This equation shows that the series converges at the certain value and we can get other function by replacing x with -x

\(\ f(x) = \frac{1}{1 + x} = \sum_{n=0}^{\infty} -x^{n} = 1 - x + x^2 + x^3 + \dots\)

The power series representation calculator helps to calculate the power series expansions of a function f(x) up to the 50 order of the series.

By keeping in view the previous condition, we suppose that a series converges as x = 0.3, then how can you prove that the series converges to a finite value? And, how does using this given x value indicate other functions? In this case, put the x value in the expression;

\(\ {\sum\limits ({x^n}}) = 1 + x + x^2 + x^3 + ...\) \(\ 1 + 0.3 + 0.09 + 0.027 + ...\)

As we add the power of variables, we reach 3 which is a finite term.

How do you Find the Power Series of a Function?

To find a power series representation for the function, write a function as an infinite series containing a variable raised to a whole number exponent. So, manually expand the series by following the steps below: 

  • Write Out the General Form
  • Determine the Coefficients
  • Substitute Coefficients into the Series
  • Expand the Series
  • Write Out the Expanded Series

Also, the online power series calculator is used for finding the series representations or checking your mathematical work. 

Example:

Let's find the power series expansion for \(\ f(x)=e^x\)

Solution:

The key to finding the power series representation of a function is its derivatives. So, evaluate the function's derivatives at x = 0.

Step # 1: Write Out the General Form

\(\ f(x) = f(a) + \frac{f'(a)(x-a)}{1!} + \frac{f''(a)(x-a)^2}{2!} + \dots + \frac{f^{n}(a)(x-a)^n}{n!}\)

Step # 2: Determine the Coefficients

We find the derivatives of f(x) that is evaluated at x = 0

\(f(x) = e^{x}\)

\(\ f'(x) = e^{x}\) derivative of \(\ e^x\) is itself

\(\ f”(x) = e^{x}\) derivative of \(\ f'(x)\) = derivative of \(\ e^x\)

\(\ f^{nx} = e^{x}\) nth derivative of \(\ e^x\) is always \(e^x\)

Therefore, the coefficients f(0), f'(0), f''(0), ... , f^(n)(0) will all be 1.

Step # 3: Substitute Coefficients into the Series

The power series for f(x) is given by:

\(\ f(x) = f(a) + \frac{f'(a)(x-a)}{1!} + \frac{f''(a)(x-a)^2}{2!} + \dots + \frac{f^{n}(a)(x-a)^n}{n!}\)

In this case, 

  • a = 0
  • f(a) = f'(a) = f''(a) = 1

Step # 4: Expand the Series

Put the values into the general form to convert function to power series:

\(\ f(x) = 1! + \frac{x}{1!} + \frac{x^2}{2!} +  \frac{x^3}{3!} + \dots + \frac{x^n}{n!}\)

Step # 5: Write Out the Expanded Series

The series can be written more concisely using summation notation:

\(\ f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}\ \text {from n = 0 to ∞}\)

This represents the sum of all the terms from n = 0 to infinity, where each term is x raised to the power of n divided by its factorial (n!). Therefore, the power series expansion from n = 0 to ∞ is:

\(\ \sum \frac{x^n}{n!}\)

Detemrining the coefficient can involves various methods that basically based on the function. However, it can be easily calculated with power series coefficient calculator.

FAQs:

Why power series converges at its center?

As we know the power series has a variable x in which the series may converge for a certain x value and diverge for others. When x equals a, the power series centered at x=a is represented by c0. It is evident in the terms that simplify to zero. Therefore a power series has convergence at its center.

What are the uses of power series?

  • Power series are used in limits
  • It is used in the evaluation of integrals
  • Power series are used in error estimation
  • Signal Processing and Filtering

Does every function have a power series?

No, not at all every function has a power series representation. These are the reasons why a function might not have a power series:

  • Discontinuity of a function
  • Infinite complexity within a function

This is how our power series calculator works to create the power series from function, it does not work if they rely on discontinuity or infinite complexity.

Can we multiply the power series?

Yes, you can multiply the power series of a function as it is just like polynomial multiplication.

Is the taylor series a power series?

Every Taylor series is a power series, but not every power series is a Taylor series. A Taylor series is always defined for a certain smooth function.

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