Integrate Power Series Calculator

A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series. Integrating a power series involves finding the antiderivative of each term in the series.

What is a Power Series?

A power series is a sum of terms where each term is a constant coefficient multiplied by a non-negative integer power of \( x \). Power series are fundamental in calculus and analysis, providing a way to represent functions as infinite sums.

Power series have a radius of convergence, which determines the interval of \( x \) values for which the series converges. The general form of a power series centered at \( c \) is:

\( \sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots \)

Power series are used to represent functions, solve differential equations, and approximate functions in numerical analysis.

How to Integrate Power Series

Integrating a power series involves finding the antiderivative of each term in the series. The integral of a power series \( \sum_{n=0}^{\infty} a_n (x - c)^n \) is another power series where each term is integrated individually.

The integral of a power series from \( a \) to \( b \) is given by:

\( \int_{a}^{b} \left( \sum_{n=0}^{\infty} a_n (x - c)^n \right) dx = \sum_{n=0}^{\infty} a_n \int_{a}^{b} (x - c)^n dx \)

For each term \( a_n (x - c)^n \), the integral is:

\( \int (x - c)^n dx = \frac{(x - c)^{n+1}}{n+1} + C \)

When integrating a power series, the constant of integration \( C \) is typically omitted for simplicity, as we're interested in the antiderivative up to a constant.

Note: The radius of convergence of the integrated series may be different from the original series. Always check the convergence of the resulting series.

Examples

Example 1: Integrating a Simple Power Series

Consider the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n+1} \). To integrate this series term by term:

\( \int \left( \sum_{n=0}^{\infty} \frac{x^n}{n+1} \right) dx = \sum_{n=0}^{\infty} \int \frac{x^n}{n+1} dx = \sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1)^2} \)

The integrated series is \( \sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1)^2} \).

Example 2: Integrating a Power Series with a Different Center

Consider the power series \( \sum_{n=0}^{\infty} n x^n \) centered at \( c = 0 \). To integrate this series:

\( \int \left( \sum_{n=0}^{\infty} n x^n \right) dx = \sum_{n=0}^{\infty} \int n x^n dx = \sum_{n=0}^{\infty} n \frac{x^{n+1}}{n+1} = \sum_{n=0}^{\infty} \frac{n x^{n+1}}{n+1} \)

The integrated series is \( \sum_{n=0}^{\infty} \frac{n x^{n+1}}{n+1} \).

FAQ

What is the radius of convergence for an integrated power series?

The radius of convergence for the integrated power series may be different from the original series. It's important to check the convergence of the resulting series.

Can all power series be integrated term by term?

Yes, power series can generally be integrated term by term within their radius of convergence. The resulting series will also converge within the same radius.

What happens if the power series does not converge?

If the power series does not converge, integrating it term by term is not valid. The series must converge for term-by-term integration to be applicable.