Convergence and divergence of integrals. Improper integrals. Signs of convergence. Absolute and conditional convergence

As is known, finding the integral can be quite difficult task. It would be a great disappointment to start calculating an improper integral and find at the end of the path that it diverges. Therefore, of interest are methods that allow, without serious calculations based on one type of function, to draw a conclusion about the convergence or divergence of an improper integral. The first and second comparison theorems, which will be discussed below, greatly help to study improper integrals for convergence.

Let f(x)?0. Then the functions

are monotonically increasing in the variables t or -g (since we take g>0, -g tends to zero from the left). If, as the arguments increase, the functions F 1 (t) and F 2 (-d) remain bounded from above, this means that the corresponding improper integrals converge. This is the basis of the first comparison theorem for integrals of non-negative functions.

Let the functions f(x) and g(x) at x?a satisfy the following conditions:

• 1) 0?f(x)?g(x);
• 2) The functions f(x) and g(x) are continuous.

Then from the convergence of the integral follows the convergence of the integral, and from the divergence of the integral follows the divergence

Since 0?f(x)?g(x) and the functions are continuous, then

By condition, the integral converges, i.e. has a finite value. Therefore, the integral also converges.

Now let the integral diverge. Let us assume that the integral converges, but then the integral must converge, which contradicts the condition. Our assumption is incorrect, the integral diverges.

Comparison theorem for improper integrals of the 2nd kind.

Let for the functions f(x) and g(x) on the interval , increase without limit for x>+0. For x>+0, the following inequality holds:<. Несобственный интеграл есть эталонный интеграл 2-го рода, который при p=<1 сходится; следовательно, по 1-й теореме сравнения для несобственных интегралов 2-го рода интеграл сходится также.

Comparison theorem for improper integrals of the 1st kind.

Let the integral diverge for the function f(x) and g(x) on the interval.

This means that the integral also diverges in the section.

Thus, this integral diverges over the entire interval [-1, 1]. Note that if we began to calculate this integral without paying attention to the discontinuity of the integrand at the point x = 0, we would get an incorrect result. Really,

, which is impossible.

So, to study the improper integral of a discontinuous function, it is necessary to “split” it into several integrals and study them.

If the integrand has a discontinuity of the second kind on the (finite) interval of integration, we speak of an improper integral of the second kind.

10.2.1 Definition and basic properties

Let us denote the integration interval by $\left[ a, \, b \right ]$; both of these numbers are assumed to be finite below. If there is only 1 discontinuity, it can be located either at point $a$, or at point $b$, or inside the interval $(a,\,b)$. Let us first consider the case when there is a discontinuity of the second kind at point $a$, and at other points the integrand function is continuous. So we are discussing the integral

\begin(equation) I=\int _a^b f(x)\,dx, (22) \label(intr2) \end(equation)

and $f(x) \rightarrow \infty $ when $x \rightarrow a+0$. As before, the first thing to do is to give meaning to this expression. To do this, consider the integral

\[ I(\epsilon)=\int _(a+\epsilon)^b f(x)\,dx. \]

Definition. Let there be a finite limit

\[ A=\lim _(\epsilon \rightarrow +0)I(\epsilon)=\lim _(\epsilon \rightarrow +0)\int _(a+\epsilon)^b f(x)\,dx. \]

Then the improper integral of the second kind (22) is said to converge and the value $A$ is assigned to it; the function $f(x)$ itself is said to be integrable on the interval $\left[ a, \, b\right]$.

Let us consider the option when there is a discontinuity of the second kind in the integrand function at the upper limit of the integration interval. This case can be reduced to the previous one by making the change of variable $x=-t$ and then rearranging the limits of integration.

Let us consider the option when the integrand function has a discontinuity of the second kind inside the integration interval, at point $c \in (a,\,b)$. In this case, the original integral

\begin(equation) I=\int _a^bf(x)\,dx (23) \label(intr3) \end(equation)

presented as a sum

\[ I=I_1+I_2, \quad I_1=\int _a^cf(x)\,dx +\int _c^df(x)\,dx. \]

Definition. If both integrals $I_1, \, I_2$ converge, then the improper integral (23) is called convergent and is assigned the value equal to the sum integrals $I_1, \, I_2$, the function $f(x)$ is called integrable on the interval $\left[a, \, b\right]$. If at least one of the integrals $I_1,\, I_2$ is divergent, the improper integral (23) is called divergent.

Convergent improper integrals of the 2nd kind have all the standard properties of ordinary definite integrals.

1. If $f(x)$, $g(x)$ are integrable on the interval $\left[ a, \,b \right ]$, then their sum $f(x)+g(x)$ is also integrable on this interval, and \[ \int _a^(b)\left(f(x)+g(x)\right)dx=\int _a^(b)f(x)dx+\int _a^(b)g (x)dx. \] 2. If $f(x)$ is integrable on the interval $\left[ a, \, b \right ]$, then for any constant $C$ the function $C\cdot f(x)$ is also integrable on this interval , and \[ \int _a^(b)C\cdot f(x)dx=C \cdot \int _a^(b)f(x)dx. \] 3. If $f(x)$ is integrable on the interval $\left[ a, \, b \right ]$, and on this interval $f(x)>0$, then \[ \int _a^(b ) f(x)dx\,>\,0. \] 4. If $f(x)$ is integrable on the interval $\left[ a, \, b \right ]$, then for any $c\in (a, \,b)$ the integrals \[ \int _a^ (c) f(x)dx, \quad \int _c^(b) f(x)dx \] also converge, and \[ \int _a^(b)f(x)dx=\int _a^(c ) f(x)dx+\int _c^(b) f(x)dx \] (additivity of the integral over the interval).

Consider the integral

\begin(equation) I=\int _0^(1)\frac(1)(x^k)\,dx. (24) \label(mod2) \end(equation)

If $k>0$, the integrand tends to $\infty$ as $x \rightarrow +0$, so the integral is improper of the second kind. Let's introduce the function

\[ I(\epsilon)=\int _(\epsilon)^(1)\frac(1)(x^k)\,dx. \]

In this case the antiderivative is known, so

\[ I(\epsilon)=\int _(\epsilon)^(1)\frac(1)(x^k)\,dx\,=\frac(x^(1-k))(1-k )|_(\epsilon)^1= \frac(1)(1-k)-\frac(\epsilon ^(1-k))(1-k). \]

for $k \neq 1$,

\[ I(\epsilon)=\int _(\epsilon)^(1)\frac(1)(x)\,dx\,=lnx|_(\epsilon)^1= -ln \epsilon. \]

for $k = 1$. Considering the behavior at $\epsilon \rightarrow +0$, we come to the conclusion that integral (20) converges at $k

10.2.2 Tests for the convergence of improper integrals of the 2nd kind

Theorem (the first sign of comparison). Let $f(x)$, $g(x)$ be continuous for $x\in (a,\,b)$, and $0 1. If the integral \[ \int _a^(b)g(x)dx \] converges, then the integral \[ \int _a^(b)f(x)dx converges. \] 2. If the integral \[ \int _a^(b)f(x)dx \] diverges, then the integral \[ \int _a^(b)g(x)dx diverges. \]

Theorem (second comparison criterion). Let $f(x)$, $g(x)$ be continuous and positive for $x\in (a,\,b)$, and let there be a finite limit

\[ \theta = \lim_(x \rightarrow a+0) \frac(f(x))(g(x)), \quad \theta \neq 0, \, +\infty. \]

Then the integrals

\[ \int _a^(b)f(x)dx, \quad \int _a^(b)g(x)dx \]

converge or diverge simultaneously.