All Questions
Tagged with natural-numbers calculus
25
questions
3
votes
1
answer
122
views
Show that $\left\{\frac{m}{mn+1}: m,n\in\Bbb N*\right\}$ is bounded
Let
$$A = \left\{\frac{m}{mn+1}: m,n\in\Bbb N*\right\}$$
The question is as follows:
Show that A has an upper bound and a lower bound and determine them.
My attempt:
I first thought of giving ...
- 39
-3
votes
1
answer
89
views
How do I solve this velocity equation? [closed]
$$t-t_0=m\int\frac{\mathrm dv}{F}=-\frac{m}{b}\int\frac{\mathrm dv}{v}$$
$$t-t_0=-\frac{m}{b}\ln\frac{v}{v_0}$$
$$v=v_0 \cdot e^{-\frac{b}{m}(t-t_0)}$$
Can you please help me? How do I convert it from ...
0
votes
2
answers
209
views
Choosing a random integer from all the natural numbers.
Context: I saw a comment that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $\infty$. Although this statement is far from rigorous this made me wonder: ...
- 6,620
-1
votes
1
answer
56
views
How can I prove the following statement and calculate the whole part of it? [closed]
I am currently trying to exercise for a competition and have the following problem for it:
$A_n = \log_{n}(n+1)+\log_{n+1}(n)$
where $n$ is natural number. I must prove that $A_n$ can never be a whole ...
- 135
2
votes
1
answer
103
views
injective sequence of natural numbers has infinite plus limit
Question
Injective sequence of natural numbers $(a_n)$ has $\lim_{n\to+\infty}a_n=+\infty$.
Draft
I thought, if it's injective, like $\mathbb N$ is not majored, it's not bounded, so it doesn't ...
- 516
0
votes
4
answers
58
views
Is there a number system base $a$ where $2018$ can be written as $\overline{21312}^{a}$ in it?
That's actually the whole question.
Is there a number system base $a$ where $2018$ can be written as $\overline{21312}^{a}$ in it?
I honestly don't know what I do more than this
$2a^4 + a^3 + 3a^2 + a ...
- 433
1
vote
1
answer
94
views
I can't prove $U_{n} \geq 1$
So we have an iteration: $U_{n+1} = 1 + \frac{1}{U_{n}+1}$ where $n$ is a natural number and $U_{0} = 1$
I have to prove $U_{n} \geq 1$ but the problem is that I just can't.
Here is my attempts:
By ...
- 433
0
votes
3
answers
71
views
Find the limit $\lim_{x \to 1}(\frac{3x}{2+x})^\frac{x}{1-x}$
Find the limit
$$
\lim_{x \to 1}\left(\frac{3x}{2+x}\right)^\frac{x}{1-x}
$$
I've transformed the function by changing limit of 1 to zero and become the following:
$$
\lim_{x \to 0}\left(1 + \frac{3x-...
- 223
2
votes
2
answers
85
views
Is there a way to integrate over reals so that the uncountable sum is finite?
I've been doing some research on this and I saw that a bounded integration over reals evaluates to infinity. This makes total sense, such as in the case:
$$
y = x
$$
If we take the typical integral ...
0
votes
1
answer
51
views
Use the notion of Integers to find a solution for $a+x=b$, $a,b,x \in \mathbb{N}$.
In my syllabus for "Scientific Computing" it states the following idea.
Consider the case $a \leq b$, $a,b \in \mathbb{N}$. This is essentially the same as $a + x = b$, $x \in \mathbb{N}$. ...
- 1,928
3
votes
3
answers
138
views
Solve in $\mathbb N^{2}$ the following equation : $5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
Question :
Solve for natural number the equation :
$5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
My try :
Let : $X=5^{x}$ and $Y=2^{y}$ so above equation
equivalent :
$2X^{2}+(...
- 2,323
4
votes
1
answer
57
views
Induction inequality related with $e$ number.
I'm stuck with induction problem. Let $0\leq a<b$, prove the next inequality for all $n\in\mathbb{N}$.
$$\frac{b^{n+1}-a^{n+1}}{b-a}<(n+1)b^n$$I need this inequality because with this, we can ...
- 3,935
-2
votes
1
answer
80
views
prove that $2^{n}+1$ is divisible by $n=3^k$ for $k≥1$ [closed]
prove that :
$2^n+1$ is divisible by all number from : $n=3^k$
for $k≥1$
I find this problems in book and I need ideas to approach it
Problems :
- 751
0
votes
2
answers
56
views
Let : $A=\frac{2^{4n+2}+1}{5}$ , $n>1$
Prove that the number A is not primary
Such that :
$A=\frac{2^{4n+2}+1}{5}$
$n≥2$
n=2 then $A=205$
Please I need some ideas to approach it
- 147
0
votes
3
answers
2k
views
Why is absolute value of negative exponent equal to positive value?
I am asked to integrate the following:
$\int_{-\infty}^{0}e^{-\left\lvert 3x\right\rvert}dx$
And I am told that $e^{-\left\lvert x\right\rvert}=e^{x}$
How is it that an absolute value (the exponent)...
- 223