# Tagged Questions

For questions about mathematical constants, that are "significantly interesting in some way".

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### Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
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### Finding constants with differentiation [on hold]

The curve y= f(x) for which f'(x)= 4x+k, where k is a constant, has a turning point at (-2, -1). a) Find the value of k. b) Find the coordinates of the point at which the curve meets the y-axis.
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### Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
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### What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ...
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### What is the value of $a + b + s + t$

I'm confused as to how to approach this. I've tried to foil it out. $$x^3(3x-1)=a+bx+sx^2+tx^3$$ The equation above is true for all values of $x$, where $a$, $b$, $s$, and $t$ are constants. What ...
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### General solution of the differential equation: y' cot x + y = 2

I have to find the general solution of the differential equation:$y$' $cot$ $x$ + $y$ = $2$. And determine the integration constant using the initial condition $y$(0) = $1$. Additionally presenting ...
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### Ways to determine $\pi$ [duplicate]

I have read that it is possible to determine the value of a single digit, say the 874th of $\pi$. I know that it is a trascentental number, how is that possible? How many ways are there to determine ...
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### A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
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### Family functions of derivatives and extremum values. [closed]

Explore the family of functions $f(x) = x^3 + kx + 1$ where $k$ is a real constant. How many and what type of local extrema are there? Your answer should depend on the value of $k$, that is, ...
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### Finding Constants on a Differential Equation

Question (from my sample exercises in calculus): Find constants $A, B$ and $C$ such that the function $$y=A\sin x+B\cos x$$ satisfies the differential equation $$y''+y'-2y=\sin x.$$ I am ...
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### Find the limit $\lim_{h\to 0}\frac{ \sqrt{7(a+h)}-\sqrt{7a} }{h}$ in terms of $a$ .

Find the limit in terms of the constant $a$ : $$\lim_{h \to 0}\frac{\sqrt{7(a+h)}-\sqrt{7a}}{h}$$ I have tried to solve this by multiplying the square roots to both sides but i simply can't solve it ...
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### Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods? UPDATE Well now I know it should not. But $e^{-\gamma}$ should.
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### Moment generating function of a constant

This might be trivial, but can you elaborate why moment generating function for a constant $c$ is $e^{cX}$, where $X$ is a random variable.
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### Solve series with two constants and opposite exponents

How can I find the general formula for the sum of this series? $$\sum_{i=0}^n a^ib^{n-i}$$ Where $a$ and $b$ are unrelated constants? I don't think you can split it into $\sum_{i=0}^n a^i$ and ...
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### Proving $\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$

How to prove that $$\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$$ Where $F_{n}$ is the Fibonacci Number.
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### Sum of infinite series - Catalan constant

Why is this identity true? $$\sum_{k=1}^{\infty} \frac{sin(k\pi/2)}{k^2} = C$$ where $C$ is Catalan constant.
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### MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper whose author has passed away. ...
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### Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?

I believe that the sum $$\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this ...
When implicitly differentiate a function, for example, $f(x)=(G)(x)$, where G is a constant, is it possible to differentiate it such that we can treat G as a variable? From my understanding it is ...
### Show that $\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$
Let $p(n)$ denote the probability that a randomly chosen one-to-one function $f:\lbrace 1,2,3...,n \rbrace \rightarrow \lbrace 1,2,3...,n \rbrace$ has a least one fixed point. (least one integer $k$ ...