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.