Power Series Calculator

Power Series Calculator:

This power series calculator allows you to expand a function into a power series for a given variable and also shows you the step-by-step calculations involved. With it, you can:

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

This calculator is very helpful for analyzing and approximating the function values.

Limitation: The calculator can handle a range of mathematical functions, but 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 formed by the multiplication of a constant coefficient (c_n) and a variable (x) raised to an increasing non-negative integer power (n), often centered around a value 'a'.”

Power Series Formula:

∑_{_{_n=0}}^{^{^∞}} c_n(x-a)ⁿ = c₀ + c₁(x-a) + c₂(x-a)² + c₃(x-a)³ + .......;

Where:

“x” is the variable
“n” is a non-negative integer that shows the power
“cn” represents constant coefficients
“a” is a constant that indicates the center of power series

When the value “a” is zero, then the series becomes simpler as:

∑_{_{_n=0}}^{^∞} c_nxⁿ = c₀ + c₁x + c₂x² + c₃x³ + ......;

This series proceeds to represent a function within the interval of convergence, and it diverges outside the interval. They are used:

To define new functions and provide alternative representations of common functions
The power series is valuable for approximating functions
Solve differential equations
Evaluate integrals

Power Series Convergence:

A power series centered at a point converges for a value of x within a certain interval. Within this interval of convergence, the absolute value of the terms typically decreases as the power increases. Hence, their sum approaches a finite value. This power series convergence can be found by using the ratio test.

∑(xⁿ) = 1 + x + x² + x³ + ... ∑_n=0^∞ xⁿ

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

At this point, its value becomes 1/(1-x). We can express the function as power series manually, as we have done below. To quickly obtain its power series representation, try our power series calculator, which takes a function as input and outputs its equivalent power series.

f(x) = 1/(1 - x) = Σ(n=0 to ∞) xⁿ = 1 + x + x² + x³ + ..., |x| < 1

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

f(x) = 1 / (1 + x) = ∑_n=0^∞ (−x)ⁿ = 1 − x + x² − x³ + … ∣−x∣<1⟹∣x∣<1

Now, let's consider the case where x=0.3. Since ∣0.3∣<1, we know that the geometric series ∑_n=0^∞(0.3)ⁿ converges. To find the finite value it converges to, we use the formula for the sum of a geometric series:

∑(xⁿ) = 1+0.3+(0.3)²+(0.3)³+⋯=1+0.3+0.09+0.027+…

The sum of this series is:

1/1−0.3 = 1/0.7 =10/7

Using x like 0.3 in the general power series gives us the numerical value that the series converges to at that point.

For advanced power series analysis, including expansions of complex functions up to a specified order, use our power series representation calculator.

How to Find the Power Series?

To find power series representation for the function, write the function as an infinite series containing a variable raised to a whole-number exponent. Follow these steps to expand the series manually:

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

Example:

Let's find the power series expansion for f(x)=eˣ

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) + (f′(a)(x − a)) / 1! + (f″(a)(x − a)²) / 2! + … + (fⁿ(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) = eˣ

f'(x) = eˣ derivative of (eˣ) is itself

f”(x) = eˣ derivative of f'(x) = derivative of eˣ

fⁿˣ = eˣ) nth derivative of (eˣ) is always (eˣ)

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 of function f(x) is given by:

f(x) = f(a) + (f′(a)(x − a)) / 1! + (f″(a)(x − a)²) / 2! + ... + (fⁿ(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 + x / 1! + x² / 2! + x³ / 3! + … + xⁿ / n!

Step # 5: Write Out the Expanded Series

The series can be written more concisely using summation notation:

f(x) = ∑_n=0^∞ xⁿ / n! from n = 0 to ∞

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

∑ xⁿ / n!

Determining the coefficient can involve various methods that are based on the function. However, it can be easily calculated with a power series coefficient calculator.

How to Use The Power Series Calculator?

Here’s how our power series calculator works to create the power series from function:

Enter the Function: Input the function exactly as it is
Specify the Variable: Choose the variable that is in your function
Input the Center Point: Provide the specific value around which the power series will be expanded
Specify the Order (n): Specify the "order" or "degree" of the series you want to form
Click on Calculate: Once you have entered all the information, click on the calculate button to perform the calculation
View the Result: The calculator will show the power series expansion of the given function up to the specified order and around the provided center point

FAQ’s:

What is a Power Series Used For?

Power series has significant applications over various fields, including:

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

Why Do Power Series Converge 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.

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

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.

References:
From the source of Wikipedia.org: Power Series.
From the source of Math.Libretexts.org: Power Series and Functions.