Convergence & Divergence

Improper integrals

Page 353

So far when working with definite integrals, they have all been defined over a finite interval. This topic will explore the cases where there is an infinite interval or functions that have an infinite discontinuity. These types of integrals are called improper.

Infinite intervals

For infinite intervals, we can split this into two possibilities:

$\displaystyle\int_{a}^{\infty} f(x) \\, dx = \displaystyle\lim_{t \to \infty} \displaystyle\int_{a}^{t} f(x) \\, dx$

$\displaystyle\int_{-\infty}^{b} f(x) \\, dx = \displaystyle\lim_{t \to -\infty} \displaystyle\int_{t}^{b} f(x) \\, dx$

In each of these cases, if the corresponding limits exist as a finite number then they are convergent and they are divergent if not. This law also covers the case of an infinite interval in both directions as:

$\displaystyle\int_{-\infty}^{\infty} f(x) \\, dx = \displaystyle\int_{-\infty}^{a} f(x) \\, dx + \displaystyle\int_{a}^{\infty} f(x) \\, dx$

Answering this kind of question involves using this fact to get a limit of a definite integral. Integrate and sub in $t$, then calculate the limit.

Infinite discontinuity

Suppose we have a positive continuous function defined on a finite interval $[a,b)$ where there is a vertical asymptote at $b$. We can write an integral of this function as:

$\displaystyle\int_{a}^{b} f(x) \\, dx = \displaystyle\lim_{t \to b^-} \displaystyle\int_{a}^{t} f(x) \\, dx$

The discontinuity can also be on the left.

$\displaystyle\int_{a}^{b} f(x) \\, dx = \displaystyle\lim_{t \to b^+} \displaystyle\int_{t}^{b} f(x) \\, dx$

Again, in these cases, if the corresponding limits exist as a finite number then they are convergent and they are divergent if not.

Effective p-series

If you get an integral of the form:

$\displaystyle\int_{1}^{\infty} \dfrac{1}{x^p} \\, dx$

Then it will converge for $p \gt 1$ and diverge for $p \le 1$.

Comparison test

If calculating the limit as above is difficult then there is another way to determine if an improper integral is convergent or divergent. Suppose $f$ and $g$ are continuous functions with $f(x) \ge g(x) \ge 0$ for $x \ge a$.

If $\int_{a}^{\infty} f(x) \\, dx$ is convergent then $\int_{a}^{\infty} g(x) \\, dx$ is also convergent.

If $\int_{a}^{\infty} g(x) \\, dx$ is divergent then $\int_{a}^{\infty} f(x) \\, dx$ is also divergent.

See the examples they use for this on page 359 of the textbook.