# All Questions

14k views

### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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### How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
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### Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$

Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
6k views

### Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. I ...
4k views

### Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
471 views

### How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
2k views

### Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
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### Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce ...
7k views

### 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 ...
8k views

### The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$? Where, \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq r\... 4answers 25k views ### What is 48÷2(9+3)? [duplicate] There is a huge debate on the internet on 48÷2(9+3). I figured if i wanted to know the answer this is the best place to ask. I believe it is 2 as i believe it is part of the bracket operation in ... 4answers 3k views ### If a \mid m and (a + 1) \mid m, prove a(a + 1) | m. Can anyone help me out here? Can't seem to find the right rules of divisibility to show this: If a \mid m and (a + 1) \mid m, then a(a + 1) \mid m. 4answers 694 views ### Prove \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} (a.k.a. Hockey-Stick Identity) [duplicate] Let n be a nonnegative integer, and k a positive integer. Could someone explain to me why the identity \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}  holds?
Let $f \in C[0,1]$ and $f(0)=f(1)$. How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$? In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$...