Power series is a type of infinite mathematical series that involves terms with a variable raised to increasing powers to infinite level. Think of it as an infinite polynomial series , which can be expressed as:

[Tex] \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \ldots[/Tex]

Here, x is the variable, and c₀, ₁, c₂, . . ., c_n​ are the coefficients. The power series provides a way to represent functions, especially those that can’t be easily written in closed form. By learning about power series, one can appreciate how infinite sums can be used to tackle problems that finite polynomials can’t handle effectively.

What is a Power Series?

Power series is a type of infinite series that represents a function as the sum of its terms. It’s a way to represent functions as infinite sums of polynomial terms. It is an infinite series that either converges or diverges. General form of power series is given as:

[Tex]\sum_{n=0}^{\infty} a_n (x – c)^n[/Tex]

Where,

• a_n: coefficient of the n^th term.
• x: variable of the series.
• c: center of the series, where the series is expanded around.
• n: term index, starting from 0 and going to infinity.

Each term of the series involves a power of (x−c). Power series can be used to approximate functions within a certain range of values for x. Understanding these series is crucial in many areas of mathematics and science.

Power Series

Examples of Power Series

Some of the common examples of power series are discussed as follows:

Example 1: Power Series of Exponential Function

The exponential function e^x can be expressed as a power series centered at x = 0 (a Maclaurin series):

[Tex] e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots [/Tex]

In this series, the coefficients a_n are [Tex]\frac{1}{n!}[/Tex], and the center c is 0.

Example 2: Power Series of Sine Function

The sine function sin(x) can also be expressed as a power series centered at x = 0:

[Tex] \sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots [/Tex]

Here, the coefficients a_n are [Tex](-1)^n \frac{1}{(2n+1)!}[/Tex], and the center c is 0.

Example 3:Power Series of Cosine Function

The cosine function cos(x) is another example of a power series centered at x = 0:

[Tex] \cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots [/Tex]

In this series, the coefficients a_n are [Tex](-1)^n \frac{1}{(2n)!}[/Tex], and the center c is 0.

Convergence of Power Series

When we talk about a converging power series, we mean that as you add more terms, the series approaches a finite value. The key concepts here are the radius of convergence and the interval of convergence.

• Radius of Convergence: The distance from the center c within which the series converges. Outside this radius, the series diverges.
• Interval of Convergence: The actual set of x-values for which the series converges. It’s centered around c and extends to the left and right by the radius of convergence.

To determine if a power series converges, we use tests like:

• Ratio Test: If [Tex]\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L[/Tex], the series converges if L<1 and diverges if L>1. When L=1, the test is inconclusive.
• Root Test: If [Tex]\lim_{n \to \infty} \sqrt[n]{|a_n|} = L[/Tex], the series converges if L<1 and diverges if L>1. When L=1, the test is inconclusive.

Radius and Interval of Convergence

Radius of convergence of a power series is a measure of the interval around the center of the series within which the series converges to a finite value.

Interval of convergence is the actual set of x-values for which the power series converges. It includes the radius of convergence and requires checking the endpoints separately. The interval is usually expressed as:

(c−R, c+R)

How to Calculate Radius of Convergence for Power Series?

We can calculate radius of convergence for any power series using

• Ratio Test
• Root Test

Using Ratio Test

To calculate radius of convergence, using ratio test we can use the following steps:

Step 1: Compute [Tex]\left| \frac{a_{n+1}}{a_n} \right|[/Tex]

Step 2: Take the limit as n approaches infinity: [Tex]L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|[/Tex]

Step 3: The radius of convergence R is given by R= 1/L if L ≠ 0.

Let’s consider example for better understanding.

Example: For the series [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!} [/Tex]

Solution:

Compute: [Tex]\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{1}{(n+1)!}}{\frac{1}{n!}} \right| = \frac{1}{n+1}[/Tex]

Take the limit: [Tex]L = \lim_{n \to \infty} \frac{1}{n+1} = 0 [/Tex]

Since L=0, the radius of convergence R = ∞.

Using Root Test

To calculate radius of convergence, using root test we can use the following steps:

Step 1: Compute [Tex]\sqrt[n]{|a_n|}[/Tex]

Step 2: Take the limit as n approaches infinity: [Tex] L = \lim_{n \to \infty} \sqrt[n]{|a_n|}[/Tex]

Step 3: The radius of convergence R is given by R= 1/L if L ≠ 0.

Let’s consider example for better understanding.

Example: For the series [Tex]\sum_{n=0}^{\infty} x^n[/Tex]

Compute: [Tex]\sqrt[n]{|a_n|} = \sqrt[n]{1} = 1[/Tex]

Take the limit: [Tex]L = \lim_{n \to \infty} 1 = 1[/Tex]

Since L=1, the radius of convergence R=1.

Operations on Power Series

Power series are flexible tools in calculus. We can perform various operations on them, such as differentiation and integration, which we’ll explore in detail.

Differentiation of Power Series

To differentiate a power series term-by-term:

• Write the general form: [Tex]\sum_{n=0}^{\infty} a_n (x – c)^n [/Tex]
• Differentiate each term with respect to x.

The derivative is:

[Tex]\frac{d}{dx} \left( \sum_{n=0}^{\infty} a_n (x – c)^n \right) = \sum_{n=1}^{\infty} n a_n (x – c)^{n-1}[/Tex]

Example:

Differentiate [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!}[/Tex]

General form: [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!}[/Tex]

Differentiate each term:

[Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!} \sum_{n=1}^{\infty} \frac{n x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}[/Tex]

Simplify to get:

[Tex]\frac{d}{dx} \left( \sum_{n=0}^{\infty} \frac{x^n}{n!} \right) = \sum_{n=0}^{\infty} \frac{x^n}{n!}[/Tex]

Integration of Power Series

To integrate a power series term-by-term:

• Write the general form: [Tex]\sum_{n=0}^{\infty} a_n (x – c)^n [/Tex]
• Integrate each term with respect to x.

The integral is:

[Tex]\int \left( \sum_{n=0}^{\infty} a_n (x – c)^n \right) dx = \sum_{n=0}^{\infty} \frac{a_n (x – c)^{n+1}}{n+1} + C[/Tex]

Example: Integrate [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!}[/Tex]

Solution:

General form: [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!}[/Tex]

Integrate each term: [Tex]\sum_{n=0}^{\infty} \frac{x^n}{n!} \sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1) n!}[/Tex]

Simplify to get:

[Tex]\int \left( \sum_{n=0}^{\infty} \frac{x^n}{n!} \right) dx = \sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1) n!} + C[/Tex]

Conclusion

Power series are a valuable tool in mathematics, allowing us to represent complex functions as an infinite sum of simpler terms. By breaking down a function into a series of powers of a variable, we can understand and work with it more easily. This method is especially useful for solving differential equations and approximating functions that are otherwise difficult to handle.

Read More,

• Integration.
• Differentiation.
• Taylor Series
• Maclurian Series
• Differential Equations

Power Series- FAQs

What is the purpose of the power series?

The intended use of power series is to express the function in terms of the infinite number of terms composed out of the powers of a certain variable.

How to tell if a series is a power series?

A series is a power series if it can be written as a sum of constants times the powers of a variable, usually x, where x can take any real or complex value.

Who discovered the power series?

Power series were explored by mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.

Are all power series geometric?

Not all power series are geometric; they can have terms that involve factorial expressions or other patterns.

Is the power series finite or infinite?

Power series are typically infinite, involving an infinite number of terms.