Improper integrals (with point at infinity)

Motivation. We will eventually talk about infinite sums in this course, and they are quite tricky. For example, consider $$1 + \frac{1}{2} + \frac{1}{3} + \cdots.$$ It looks like each time we add something smaller than the previous term, but in fact, we have $$1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty.$$ How do we know? First, do the following exercise:

Exercise. By drawing a good picture, explain that $$1 + \frac{1}{2} + \frac{1}{3} + \cdots \geq \int_{1}^{\infty}\frac{1}{x}dx.$$ (Hint: the numbers we add will be the areas of certain boxes.)

Now, we have $$\begin{align*} \int_{1}^{\infty}\frac{1}{x}dx &= \lim_{r \rightarrow \infty}\int_{1}^{r}\frac{1}{x}dx \\ &= \lim_{r \rightarrow \infty}\left.\ln(|x|)\right|_{1}^{r} \\&= \lim_{r \rightarrow \infty}\ln(r) \\&= \infty. \end{align*}$$ This lets us see that the infinite sum is infinite.

Exercise. Determine whether the following sum is finite or infinite: $$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots.$$ This will be a part of somewhat general behavior:

Exercise. Figure out exactly which real number $p$ makes the following integral finite: $$\int_{1}^{\infty}\frac{1}{x^{p}}dx.$$ The finite integrals will be called converegent, while the infinite integrals will be called divergent.

Exercise. Read Section 7.6.

Exercise. Determine whether the following integral is convergent or divergent: $$\int_{1}^{\infty} \frac{1}{(x+4)^{2}}dx.$$ (Hint: Use substitution trick.)

Exercise. Determine whether the following integral is convergent or divergent: $$\int_{1}^{\infty} \frac{x}{(x+4)^{2}}dx.$$ (Hint: Use substitution trick and then split the integral into two.)

Exponential function. One improper integral involving $\infty$ that comes up a lot is the exponential decay. Namely, the function $e^{-x}$ decays so fast that $$\int_{1}^{\infty} e^{-x}dx$$ is finite.

Exercise. Explain why $$\frac{1}{x^{2}} \leq e^{-x}$$ for large enough $x.$ Using this information, show that the following integral is finite: $$\int_{1}^{\infty} e^{-x}dx.$$ (Hint: Try to think about using L'H on $x^{2}/e^{x}.$)

Exercise. Compute $$\int_{1}^{\infty} e^{-x}dx$$ as an exact value.

Exercise. Figure out which $a$ makes the following integral finite: $$\int_{1}^{\infty} e^{ax}dx.$$ (Hint: For $a \geq 0,$ note that $e^{ax} \geq 1$ for all $x.$)

Exercise. Determine the convergence of the following integral: $$\int_{3}^{\infty} x^{3}e^{-x}dx.$$ (Hint: Recall that $e^{x} \geq x^{5}$ for large enough $x.$ Why does this help?)

Exercise. Determine the convergence of the following integral: $$\int_{3}^{\infty} \ln(x)(e^{-x} + x^{-1})dx.$$ (Hint: Try to split the integral.)

Exercise. Determine the convergence of the following integral: $$\int_{3}^{\infty} x^{-3000}e^{x}dx.$$ (Hint: $e^{x} \geq x^{3000}$ eventually in $x$.)

Exercise. Determine the convergence of the following integral: $$\int_{3}^{\infty} \sin^{2}(x)x^{-2}dx.$$ (Hint: $-1 \leq \sin(x) \leq 1$.)