Tag: Improper Integrals

Prove or disprove: ∫ f(x) converges implies lim f(x) = 0

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

The convergence of the improper integral

$\int_1^{\infty} f(x) \, dx$

implies

$\lim_{x \to \infty} f(x) = 0.$

Counterexample. The idea of the construction is a function which has rapidly diminishing area, but has a height that is not going to 0. (So, for an idea consider triangles on the real line all with height 1, but for which the base is becoming small rapidly.) To make this concrete, define

$f(x) = 1 - 2n^2 \left| x - \left( n + \frac{1}{2n^2} \right) \qquad \text{for } x \in [n,n+1),$

for each positive integer $n$ . Then for the improper integral we have

$\int_1^{\infty} f(x) \, dx \leq \sum_{n=1}^{\infty} \frac{1}{2n^2}$

which we know converges. On the other hand

$f \left( n + \frac{1}{2n^2} \right) = 1$

for all positive integers $n$ . Hence,

$\lim_{x \to \infty} f(x) \neq 0$

(since it does not exist). Hence, the statement is false.

(Note: For more on this see this question on Math.SE.)

Prove or disprove a statement relating the derivative of a function to an improper integral of the function

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

Assume $f'(x)$ exists for all $x \geq 1$ and is bounded,

$|f'(x)| \leq M$

for some constant $M$ for all $x \geq 1$ . Then,

$\lim_{n \to \infty} I_n = A \quad \implies \quad \int_1^{\infty} f(x) \, dx = A.$

Incomplete.

Prove or disprove: If f is positive and lim I_n = A, then ∫ f(x) converges to A

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

If $f$ is positive and if

$\lim_{n \to \infty} I_n = A$

then

$\int_1^{\infty} f(x) \, dx = A.$

Incomplete.

Prove or disprove: If the sequence I_n converges, then ∫ f(x) converges

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

The convergence of the sequence $\{ I_n \}$ implies the convergence of the integral

$\int_1^{\infty} f(x) \, dx.$

Incomplete.

Prove or disprove: If lim f(x) = 0 and lim I_n = A then ∫ f(x) converges to A

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

Assume

$\lim_{x \to \infty} f(x) = 0 \qquad \text{and} \qquad \lim_{n \to \infty} I_n = A.$

Then

$\int_1^{\infty} f(x) \, dx = A.$

Incomplete.

Prove or disprove: If f is monotonic decreasing and lim I_n exists then ∫ f(x) converges

by RoRi

March 31, 2016

The following function $f$ is defined for all $x \geq 1$ , and $n$ is a positive integer. Prove or provide a counterexample to the following statement.

If $f$ is monotonically decreasing and if

$\lim_{n \to \infty} I_n$

exists, then the improper integral

$\int_1^{\infty} f(x) \, dx$

converges.

Incomplete.

Use integration by parts to prove the functional equation of the Gamma function

by RoRi

March 31, 2016

Recall the definition of the Gamma function:

$\Gamma(s) = \int_{0^+}^{\infty} t^{s-1} e^{-t} \, dt \qquad \text{for} \quad s > 0.$

Using integration by parts, prove the functional equation:

$\Gamma(s+1) = s \Gamma(s).$

Then use mathematical induction to prove that for positive integers $n$ we have

$\Gamma(n+1) = n!.$

Incomplete.

Prove that the improper integral ∫ f(x) dx and ∑ f(n) both converge or both diverge

by RoRi

March 31, 2016

Assume that $f$ is a monotonically decreasing function for all $x \geq 1$ and that
$\lim_{x \to +\infty} f(x) = 0.$

Prove that the improper integral and the series

$\int_1^{\infty} f(x) \, dx \qquad \text{and} \qquad \sum_{n=1}^{\infty} f(n)$

both converge or both diverge.
Give a counterexample to the theorem in part (a) in the case that $f$ is not monotonic, i.e., find a non-monotonic function $f$ such that $\sum f(n)$ converges but $\int_1^{\infty} f(x) \, dx$ diverges.

Incomplete.

Prove some properties of the improper integrals ∫ (sin x) / x and ∫ (cos t) / t²

by RoRi

March 31, 2016

Prove that the following improper integral converges:
$\int_{0^+}^1 \frac{\sin x}{x} \, dx.$
Prove that
$\lim_{x \to 0^+} x \int_x^1 \frac{\cos t}{t^2} \, dt = 1.$
Determine the convergence or divergence of the improper integral
$\int_{0^+}^1 \frac{\cos t}{t^2} \, dt.$

Incomplete.

Prove some properties of improper integrals involving 1/x and sin x

by RoRi

March 31, 2016

Prove the following limit formulas:
$\lim_{h \to 0^+} \left( \int_{-1}^{-h} \frac{dx}{x} + \int_h^1 \frac{dx}{x} \right) = 0, \qquad \lim_{h \to +\infty} \int_{-h}^h \sin x \, dx = 0.$
Determine whether the following improper integrals converge:
$\int_{-1}^1 \frac{dx}{x}; \qquad \int_{-\infty}^{\infty} \sin x \, dx.$

Incomplete.