Power Series Calculator

Power series Calculator 

Convert a function into the power series expansion using this power series representation calculator with steps. It uses the correct formula to formulate the series and can find up to the 10th order of the series. 

What is power series?

A power series is a mathematical tool used to represent and analyze functions. It is a series (a sum of terms) where each term is a constant multiplied by a variable raised to increasing power. 

In simple words, it's like adding up terms that have a pattern, where the pattern involves a number (the variable) being multiplied by itself an increasing number of times. 

Power series formula

The general form of a power series can be written as:

Σ(a_n * x_n) = a₀ + a₁x + a₂x² + a₃x³ + ...

Here,

• Σ denotes the summation symbol (which means to add up an infinite number of terms),
• 'a_n' represents the coefficients (which can be any real or complex number),
• 'x' is the variable,
• 'n' is the exponent, ranging from 0 to infinity. 

The series converges within a specific range of x values called the radius of convergence.

Power series convergence

Series convergence means that the sum of its terms approaches a finite value as the number of terms increases toward infinity.

In other words, as more and more terms are added in the series, the total sum doesn't go to infinity or oscillate indefinitely but rather settles at a specific number.

For example, consider the geometric series:

Σ(x_n) = 1 + x + x² + x³ + ...

This series converges when the absolute value of x is less than 1 (|x| < 1). If we pick a value for x that meets this condition, say x = 0.5, the series becomes:

1 + 0.5 + 0.52 + 0.53 + ... = 1 + 0.5 + 0.25 + 0.125 + ...

As we add more and more terms to the sum, the series converges to 2, which is the finite limit. In this case, the sum of the infinite terms doesn't go to infinity but rather approaches the finite value 2.

How to expand the power series?

To find the power series of a given function, express the function as an infinite sum of terms involving a variable raised to a whole number power. 

This can be done by using known power series representations of functions and manipulating them using algebraic operations like addition, multiplication, and differentiation.

Here are a few examples of finding power series:

Example 1

Find the power series for f(x) = sin(x).

Solution:

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

sin(x) = Σ((-1)n * x²ⁿ⁺¹ / (2n+1)!) = x - x³/3! + x⁵/5! - x⁷/7! + ...

To derive this series, we start with the Taylor series expansion of e^ix:

e^ix = cos(x) + i sin(x)

We can rearrange this equation to solve for sin(x):

sin(x) = Im(e^ix) = (1/i)(e^ix - e^-ix)

Now, we can substitute the Taylor series expansions of e^ix and e-ix into this expression and simplify to obtain the power series for sin(x).

Example 2

Find the power series for f(x) = 1/(1+x).

Solution:

We can start by using the geometric series:

Σ(xⁿ) = 1 + x + x² + x³ + ...

If we multiply this series by (1-x), we get:

(1-x)Σ(xⁿ) = (1-x) + (x-x²) + (x²-x³) + (x³-x⁴) + ...

Simplifying this expression, we get:

(1-x)Σ(xⁿ) = 1

Dividing both sides by (1-x), we obtain:

Σ(xⁿ) = 1/(1-x)

Now, we can substitute x = -x in this expression to get the power series for 1/(1+x):

Σ((-1)ⁿ * xⁿ) = 1 - x + x² - x³ + ...

Use of power series

Differential equations: Power series can be used to solve differential equations by assuming that the solution can be represented as a power series. It is because power functions are the easiest when it comes to differentiating. So, with power series, a function is represented  This technique is called the method of power series.

Approximation of functions: Power series can be used to represent many common functions such as the exponential function, trigonometric functions, and logarithmic functions. We can obtain an approximate function value for a given input using a finite number of terms from the series.

Signal processing: Power series are used to represent signals as a sum of weighted sinusoids. This is the basis for the Fourier series, which is used extensively in signal-processing applications.