Tag: Limits

Find the angle between the vectors (2,4,6, …, 2n) and (1,3,5,…,2n-1) as n goes to infinity

by RoRi

April 28, 2016

Consider the vectors $A = (2,4,6, \ldots, 2n)$ and $B = (1,3,5, \ldots, 2n-1)$ in $\mathbb{R}^n$ . Let $\theta$ denote the angle between $A$ and $B$ . Find the value of $\theta$ as $n \to \infty$ .

If $\theta$ is the angle between $A$ and $B$ then we have

$\cos \theta &= \frac{A \cdot B}{\lVert A \rVert \lVert B \rVert}.$

Computing each of these pieces we have

$\begin{align*} A \cdot B &= \sum_{k=1}^n (2k)(2k-1) \\[9pt] &= \sum_{k=1}^n (4k^2 - 2k) \\[9pt] &= 4 \left( \frac{2n^3 + 3n^2 + n}{6} \right) - 2 \left( \frac{n^2+n}{2} \right) \\[9pt] &= \frac{4n^3 + 3n^2 - n}{3}.\\[9pt] \lVert A \rVert &= \left( \sum_{k=1}^n (2k)^2 \right)^{\frac{1}{2}} \\[9pt] &= \left( 4 \left( \frac{2n^3+3n^2 + n}{6} \right) \right)^{\frac{1}{2}} \\[9pt] &= \left( \frac{4n^3 + 6n^2 + 2n}{3} \right)^{\frac{1}{2}}.\\[9pt] \lVert B \rVert &= \left(\sum_{k=1}^n (2k-1)^2 \right)^{\frac{1}{2}} \\[9pt] &= \left( \sum_{k=1}^n 4k^2 - \sum_{k=1}^n 4k + \sum_{k=1}^n 1 \right)^{\frac{1}{2}} \\[9pt] &= \left( \frac{4n^3+3n^2+2n}{3} \right)^{\frac{1}{2}}. \end{align*}$

Therefore,

$\cos \theta = \frac{A \cdot B}{\lVert A \rVert \lVert B \rVert} = \left( \frac{4n^3+3n^2-n}{3} \right) \left( \frac{9}{16n^3 + o(n^2)} \right)^{\frac{1}{2}}.$

Hence, $\cos \theta \to 1$ as $n \to \infty$ which implies $\theta \to 0$ as $n \to \infty$ .

Find the angle between the vectors (1,1, …, 1) and (1,2,…,n) as n goes to infinity

by RoRi

April 28, 2016

Consider the vectors $A = (1,1, \ldots, 1)$ and $B = (1,2, \ldots, n)$ in $\mathbb{R}^n$ . Let $\theta$ denote the angle between $A$ and $B$ . Find the value of $\theta$ as $n \to \infty$ .

If $\theta$ is the angle between $A$ and $B$ then we have

$\begin{align*} \cos \theta &= \frac{A \cdot B}{\lVert A \rVert \lVert B \rVert} \\[9pt] &= \frac{\sum_{k=1}^n k}{\sqrt{n} \sqrt{\sum_{k=1}^n k^2}} \\[9pt] &= \frac{\frac{n(n+1)}{2}}{\sqrt{n} \sqrt{ \frac{2n^3+3n^2+n}{6} }} \\[9pt] &= \frac{\sqrt{3}}{\sqrt{2}} \left( \frac{n^2 + n}{\sqrt{2n^4 + 3n^3 + n^2}} \right). \end{align*}$

Thus,

$\lim_{n \to \infty} \cos \theta = \lim_{n \to \infty} \frac{\sqrt{3}}{\sqrt{2}} \left( \frac{n^2 + n}{\sqrt{2n^4 + 3n^3 + n^2}} \right) = \frac{\sqrt{3}}{2}.$

Therefore, $\theta \to \frac{\pi}{6}$ as $n \to \infty$ .

Show that the limit and derivative cannot be interchanged when f_n(x) = (sin (nx)) / n

by RoRi

March 31, 2016

For each positive integer $n$ and and all real $x$ define

$f_n (x) = \frac{\sin (nx)}{n}, \qquad f(x) = \lim_{n \to \infty} f_n(x).$

Prove that

$\lim_{n \to \infty} f_n'(0) \neq f'(0).$

Proof. First, we have

$f_n (x) = \frac{\sin (nx)}{n} \quad \implies \quad f_n'(x) = \cos (nx) \quad \implies \quad f_n'(0) = 1$

for all $n$ . Hence,

$\lim_{n \to \infty} f_n'(0) = 1.$

On the other hand,

$f(x) = \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{\sin (nx)}{n} = 0 \implies f'(0) = 0.$

Hence,

$\lim_{n \to \infty} f_n'(0) \neq f'(0). \qquad \blacksquare$

Show that we cannot interchange a limit and integral for f_n(x) = nxe^-nx²

by RoRi

March 31, 2016

For each positive integer $n$ define

$f_n (x) = nxe^{-nx^2} \qquad x \in \mathbb{R}.$

Prove that the following limit and integral cannot be interchange:

$\lim_{n \to \infty} \int_0^1 f_n (x) \, dx \neq \int_0^1 \left( \lim_{n \to \infty} f_n (x) \, dx \right).$

Proof. First, we have

$\begin{align*} \lim_{n \to \infty} \left( \int_0^1 f_n (x) \, dx \right) &= \lim_{n \to \infty} \left( \int_0^1 nxe^{-nx^2} \, dx \right) \\[9pt] &= \lim_{n \to \infty} \left( -\frac{1}{2} e^{-nx^2} \Bigr \rvert_0^1 \right) \\[9pt] &= \lim_{n \to \infty} \left( -\frac{1}{2} \left( e^{-n} - 1 \right) \right) \\[9pt] &= \frac{1}{2}. \end{align*}$

On the other hand,

$\begin{align*} \int_0^1 \left( \lim_{n \to \infty} f_n (x) \right) \, dx &= \int_0^1 \left( \lim_{n \to \infty} nxe^{-nx^2} \right) \, dx \\[9pt] &= \int_0^1 0 \, dx \\[9pt] &= 0. \end{align*}$

Hence,

$\lim_{n \to \infty} \int_0^1 f_n (x) \, dx \neq \int_0^1 \lim_{n \to \infty} f_n (x) \, dx. \qquad \blacksquare$

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 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.

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.