# All Questions

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### Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$

Could someone explain to me why the identity $$\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$$ holds?
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### Discrete logarithm tables for the fields $\Bbb{F}_8$ and $\Bbb{F}_{16}$.

The smallest non-trivial finite field of characteristic two is $$\Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\},$$ where $\beta$ and $\beta+1$ are primitive cubic roots of unity, and zeros of the ...
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### Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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### How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
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### Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
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### Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
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### Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
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### Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
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### Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
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### Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
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### Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
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### Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$

Consider the sequence defined as $x_1 = 1$ $x_{n+1} = \sin x_n$ I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
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### In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: ...
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### Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
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### Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means

Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: ...
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### Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
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### Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In ...
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### Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
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### Why 1 is not considered to be a prime number?

Why $1$ is not considered to be a prime number? Or why definition of prime numbers is given for integers greater than $1$?
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### $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $\displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
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### Classification of prime ideals of $\mathbb{Z}[X]$
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? (1) $(0)$ (2) ...