# All Questions

15answers
9k 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 ...
12answers
20k 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
13k 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 ...
22answers
22k views

### Is $.\overline{9} = 1$?

I'm told by smart people that $.\overline{9} = 1$ and I believe them, but is there a proof that explains why this is?
16answers
43k views

### Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
4answers
8k views

### How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
25answers
33k views

### Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
3answers
538 views

4answers
2k 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 $a(a + 1) \mid m$.
10answers
3k views

### Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

We know $$i^2=-1$$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 ...
3answers
2k views

So I have been trying for a few days to figure out the sum of $$S = \sum_{k=1}^\infty \frac{1}{k^2 - a^2}$$ where $a \in (0,1)$. So far from my nummerical analysis and CAS that this sum equals $$... 5answers 8k views ### How to define a bijection between (0,1) and (0,1]? How to define a bijection between (0,1) and (0,1]? Or any other open and closed intervals? If the intervals are both open like (-1,2)\text{ and }(-5,4) I do a cheap trick (don't know if ... 5answers 5k views ### Highest power of a prime p dividing N! How does one find the highest power of a prime p that divides N! and other related products? Related question: How many zeros are there at the end of N!? This is being done to reduce ... 6answers 2k views ### Linear diophantine equation 100x - 23y = -19 I need help with this equation:$$100x - 23y = -19.$$When I plug this into Wolfram|Alpha, one of the integer solutions is x = 23n + 12 where n is a subset of all the integers, but I can't seem ... 5answers 4k views ### Finding the limit of \frac {n}{\sqrt[n]{n!}} I'm trying to find$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling. 6answers 4k views ### x^y = y^x for integers x and y We know that 2^4 = 4^2 and (-2)^{-4} = (-4)^{-2}. Is there another pair of integers x, y (x\neq y) which satisfies the equality x^y = y^x? 5answers 3k views ### Limits: How to evaluate \lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x What methods can be used to evaluate the limit$$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$In other words, if I am given a polynomial P(x)=x^n + a_{n-1}x^{n-1} ... 12answers 9k views ### Prove that \lim \limits_{n \to \infty} \frac{x^n}{n!} = 0, x \in \Bbb R. Why is$$\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 ... 10answers 37k views ### What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ... 4answers 14k 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} ... 7answers 8k 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 ... 2answers 3k views ### Convergence a.e. and of norms implies that in Lebesgue space 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 is it ... 2answers 21k 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 ... 34answers 20k views ### Examples of apparent patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ... 2answers 9k 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, ... 1answer 372 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 ...

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