# All Questions

16answers
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 ...
15answers
26k views

### Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
16answers
29k views

### 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 ...
21answers
28k views

### Is $0.999999999\ldots = 1$?

I'm told by smart people that $0.999999999\ldots = 1$, and I believe them, but is there a proof that explains why this is?
5answers
7k views

5answers
13k views

### How to show that $f(x)=x^2$ is continuous at $x=1$? [closed]

How to show that $f(x)=x^2$ is continuous at $x=1$?
2answers
32k views

### Expected time to roll all 1 through 6 on a die

What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained ...
8answers
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 ...
3answers
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 ...
2answers
4k views

### If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why ...
6answers
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.
9answers
3k views

### Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: ...
3answers
4k views

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 ... 4answers 21k views ### Norms Induced by Inner Products and the Parallelogram Law Let  V  be a normed vector space (over \mathbb{R}, say, for simplicity) with norm  \lVert\cdot\rVert. It's not hard to show that if \lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle} ... 2answers 15k views ### Limit of L^p norm Could someone help me prove that given a finite measure space (X, \mathcal{M}, \sigma) and a measurable function f:X\to\mathbb{R} in L^\infty and some L^q, ... 5answers 3k views ### Limits: How to evaluate \lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates. What methods can be used to evaluate the limit ... 5answers 8k views ### 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 ... 10answers 3k views ### Why \sqrt{-1 \times -1} \neq \sqrt{-1}^2? [duplicate] We knowi^2=-1 $$then why does this happen?$$ i^2 = \sqrt{-1}\times\sqrt{-1}  =\sqrt{-1\times-1}  =\sqrt{1}  = 1 $$EDIT: I see this has been dealt with before but at least with ... 1answer 630 views ### Discrete logarithm tables for the fields \Bbb{F}_8 and \Bbb{F}_{16}. The smallest non-trivial finite field of characteristic two is$$ \Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\}, $$where \beta and \beta+1 are primitive cubic roots of unity, and zeros of the ... 13answers 16k views ### Prove that \lim \limits_{n \to \infty} \frac{x^n}{n!} = 0, x \in \Bbb R. Why is$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$Can we generalize it to any exponent x \in \Bbb R? This is to say, is$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$This is ... 6answers 6k views ### x^y = y^x for integers x and y We know that 2^4 = 4^2 and (-2)^{-4} = (-4)^{-2}. Is there another pair of integers x, y (x\neq y) which satisfies the equality x^y = y^x? 7answers 15k views ### Show that the set of all finite subsets of \mathbb{N} is countable. Show that the set of all finite subsets of \mathbb{N} is countable. I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the ... 3answers 2k views ### Universal Chord Theorem Let f \in C[0,1] and f(0)=f(1). How do we prove \exists a \in [0,1/2] such that f(a)=f(a+1/2)? In fact, for every positive integer n, there is some a, such that f(a) = ... 9answers 12k views ### Is there an elementary proof that \sum \limits_{k=1}^n \frac1k is never an integer? If n>1 is an integer, then \sum \limits_{k=1}^n \frac1k is not an integer. If you know Bertrand's Postulate, then you know there must be a prime p between n/2 and n, so \frac 1p ... 2answers 3k views ### Prove convergence of the sequence (z_1+z_2+\cdots + z_n)/n of Cesaro means Prove that if \lim_{n \to \infty}z_{n}=A then:$$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A I was thinking spliting it in: ...

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