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Power Series

Last Updated : 23 Jul, 2025

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:

Here, x is the variable, and c0, 1, c2, . . ., cn​ are the coefficients.

General Form of Power Series

General form of power series is given as:

Where,

  • an: coefficient of the nth 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.

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 ex can be expressed as a power series centered at x = 0 (a Maclaurin series):

In this series, the coefficients an are , 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:

Here, the coefficients an are , 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:

In this series, the coefficients an are , 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. The interval is usually expressed as:

(c−R, c+R)

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

  • Ratio Test: If , the series converges if L<1 and diverges if L>1. When L=1, the test is inconclusive.
  • Root Test: If , the series converges if L<1 and diverges if L>1. When L=1, the test is inconclusive.

Let's consider examples for better understanding.

Example: For the series

Solution:

Compute:

Take the limit:

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

Example: For the series .

Solution:

Compute:

Take the limit:

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:
  • Differentiate each term with respect to x.

The derivative is:

Example:

Differentiate

General form:

Differentiate each term:

Simplify to get:

Integration of Power Series

To integrate a power series term-by-term:

  • Write the general form:
  • Integrate each term with respect to x.

The integral is:

Example: Integrate

Solution:

General form:

Integrate each term:

Simplify to get:

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