In mathematics, an improper integral is an integral over an infinite interval or over a function that has an infinite discontinuity. Improper integrals are often used to calculate the area under a curve, the volume of a solid, or the length of a curve.
One way to determine whether an improper integral diverges is to use series expansion. Series expansion is a method for representing a function as a sum of simpler functions. By expanding the function in a series, we can often determine whether the integral of the function diverges.
Example
Consider the following improper integral:
$$\int_1^\infty \frac{1}{x^2} dx$$
We can expand the function $\frac{1}{x^2}$ in a series as follows:
$$\frac{1}{x^2} = \frac{1}{1} – \frac{1}{x} + \frac{1}{x^2} – \frac{1}{x^3} + \cdots$$
Integrating each term of the series, we get:
$$\int_1^\infty \frac{1}{x^2} dx = \int_1^\infty \left(\frac{1}{1} – \frac{1}{x} + \frac{1}{x^2} – \frac{1}{x^3} + \cdots\right) dx$$
$$= \left[x – \ln|x| – \frac{1}{x} + \frac{1}{2x^2} – \cdots\right]_1^\infty$$
$$= \lim_{x\to\infty} \left[x – \ln|x| – \frac{1}{x} + \frac{1}{2x^2} – \cdots\right] – \left[1 – \ln|1| – \frac{1}{1} + \frac{1}{2(1)^2} – \cdots\right]$$
$$= \lim_{x\to\infty} \left(x – \ln|x| – \frac{1}{x} + \frac{1}{2x^2} – \cdots\right) – 0$$
$$= \infty$$
Since the limit of the integral is infinite, the improper integral diverges.
Applications
Series expansion can be used to prove the divergence of a wide variety of improper integrals. For example, series expansion can be used to prove the divergence of the following integrals:
- $$\int_1^\infty \frac{1}{x^p} dx$$ where $p \leq 1$
- $$\int_0^\infty e^{-x^2} dx$$
- $$\int_0^\infty \sin x dx$$
Series expansion can also be used to prove the convergence of some improper integrals. For example, series expansion can be used to prove the convergence of the following integrals:
- $$\int_1^\infty \frac{1}{x^p} dx$$ where $p > 1$
- $$\int_0^\infty e^{-x} dx$$
- $$\int_0^\infty \frac{1}{x} e^{-x} dx$$
Conclusion
Series expansion is a powerful tool for determining the convergence or divergence of improper integrals. By expanding the function in a series, we can often determine whether the integral of the function converges or diverges.