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### 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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### 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?
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### Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
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### Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix ...
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### How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly ...
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### Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
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### The staircase paradox, or why $\pi\ne4$

What is wrong with this proof? Is $\pi=4?$
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### Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
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### How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/...
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### Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can ...
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### Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is known as "The sum of the first $n$ positive ...
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### How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to ...
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It's stated that the gradient of: $$\frac{1}{2}x^TAx - b^Tx +c$$ is $$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$ How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^... • 733 172 votes 17 answers 151k views ### How to prove Euler's formula:$e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? Could you provide a proof of Euler's formula:$e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? • 7,620 148 votes 2 answers 42k views ### Examples of bijective map from$\mathbb{R}^3\rightarrow \mathbb{R}$Could any one give an example of a bijective map from$\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you. • 31.3k 138 votes 32 answers 75k views ### Sum of First$n$Squares Equals$\frac{n(n+1)(2n+1)}{6}$I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ... • 1,733 48 votes 3 answers 53k views ### Explain why$E(X) = \int_0^\infty (1-F_X (t)) \, dt$for every nonnegative random variable$X$Let$X$be a non-negative random variable and$F_{X}$the corresponding CDF. Show, $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$ when$X$has : a) a discrete distribution, b) a continuous ... • 1,371 83 votes 15 answers 86k 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 being ... • 1,175 45 votes 8 answers 21k views ### Why is$a^n - b^n$divisible by$a-b$? I did some mathematical induction problems on divisibility$9^n-2^n$is divisible by 7.$4^n-1$is divisible by 3.$9^n-4^n$is divisible by 5. Can these be generalized as$a^n-$... • 557 251 votes 9 answers 29k 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 WolframAlpha, I might guess that the limit is$\frac{1}{2}$, which is a pretty interesting and nice ... • 43.2k 124 votes 23 answers 24k views ### 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: $$\begin{array}{|c|c|c|} \... • 1,359 29 votes 1 answer 4k views ### Discrete logarithm tables for the fields \Bbb{F}_8 and \Bbb{F}_{16}. The smallest non-trivial finite field of characteristic two is$$ \Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\}, $$where \beta and \beta+1 are primitive cubic roots of unity, and zeros of the ... 17 votes 1 answer 9k views ### \sum \cos when angles are in arithmetic progression [duplicate] Possible Duplicate: How can we sum up \sin and \cos series when the angles are in arithmetic progression? Prove$$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\... • 179 171 votes 2 answers 96k views ### Discontinuous derivative. [duplicate] Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-... • 1,827 153 votes 3 answers 23k 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 elementary ... • 1,633 270 votes 1 answer 24k views ### 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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