User RRL

46 votes

The math behind Warren Buffett's famous rule – never lose money

answered Dec 7, 2018 at 18:12

32 votes

Prove $\lim_{n \to \infty} \frac{\ln(n)}{n}=0$ without L'Hospital's Rule

calculus sequences-and-series limits limits-without-lhopital

answered Feb 6, 2016 at 4:51

27 votes

Integrability of Thomae's Function on $[0,1]$.

real-analysis integration analysis proof-verification riemann-integration

answered Sep 9, 2015 at 8:03

20 votes

Accepted

A nonnegative random variable has zero expectation if and only if it is zero almost surely

probability-theory measure-theory

answered Aug 15, 2014 at 2:25

18 votes

Why do some mathematical ideas seem counter-intuitive?

soft-question intuition

answered Oct 10, 2014 at 4:49

18 votes

Proving that $\gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

calculus integration summation improper-integrals

answered Jun 22, 2014 at 23:31

17 votes

Accepted

Proof that if $f$ is integrable then also $f^2$ is integrable

real-analysis integration

answered Oct 13, 2014 at 23:03

17 votes

Geometric mean of reals between 0 and 1

means

answered Oct 30, 2014 at 4:02

17 votes

Accepted

How to compute the pade approximation?

numerical-methods

answered Jul 8, 2014 at 17:03

16 votes

Accepted

Uncorrelated but not independent random variables

probability random-variables correlation

answered Apr 1, 2015 at 4:58

15 votes

How to evaluate the sum $\sum_{k=2}^{\infty}\log{(1-1/k^2)}$?

sequences-and-series

answered Feb 28, 2015 at 7:46

15 votes

How to show that $\lim_{n\to \infty} \frac{a_1 +a_2 + \cdots + a_n}{n} = 0?$

real-analysis sequences-and-series limits solution-verification alternative-proof

answered Jul 2, 2021 at 21:50

15 votes

Accepted

Convergence $ \sum \frac{\sin \sqrt{n}}{n^{3/2}}, \quad \sum \frac{\sin n}{\sqrt{n}}, \quad\sum \frac{\sin \sqrt{n}}{n^{3/4}} $

calculus sequences-and-series

answered Dec 22, 2016 at 19:14

15 votes

Accepted

Prove that $\lim\limits_{n\rightarrow\infty} \left(1+\frac{1}{a_{n}} \right)^{a_{n}}=e$ if $\lim\limits_{n\rightarrow\infty} a_{n}=\infty$

calculus sequences-and-series limits

answered Dec 6, 2017 at 17:21

14 votes

Accepted

Numerical Integration for integrable singularity

integration numerical-methods complex-integration simpsons-rule

answered Mar 14, 2017 at 23:51

14 votes

On the convergence of $\sum_{n = 1}^\infty\frac{\sin\left(n^a\right)}{n^b}$

calculus sequences-and-series convergence-divergence divergent-series

answered Jul 28, 2017 at 22:43

13 votes

Can someone explain in general what a central difference formula is and what it is used for?

numerical-methods taylor-expansion approximation

answered Aug 5, 2014 at 16:15

13 votes

Accepted

Proof of connection between improper Riemann Integral and Lebesgue integral

real-analysis integration improper-integrals lebesgue-integral

answered Jul 30, 2017 at 7:49

12 votes

Accepted

Evaluate the limit $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}\right)$

sequences-and-series limits

answered Jun 6, 2016 at 3:45

12 votes

Accepted

Evaluating a Lebesgue Integral

real-analysis integration definite-integrals lebesgue-integral

answered Nov 6, 2019 at 18:48

12 votes

Deriving Taylor series without applying Taylor's theorem.

calculus real-analysis taylor-expansion recreational-mathematics

answered Nov 9, 2016 at 0:07

12 votes

Accepted

Riemann-Stieltjes integral of unbounded function

real-analysis integration analysis riemann-integration stieltjes-integral

answered Jun 5, 2017 at 19:51

12 votes

Accepted

Is $\sum_{n=2}^{\infty}\frac{(-1)^n}{\sqrt n+(-1)^n}$ convergent?

sequences-and-series analysis proof-verification alternative-proof

answered Sep 9, 2017 at 21:09

12 votes

Accepted

Proof every convex function is continuous (Problem 10 Convex Functions Spivak)

real-analysis proof-verification convex-analysis

answered Feb 22, 2018 at 21:00

12 votes

Riemann integral on a single point

calculus integration riemann-integration

answered Jul 30, 2018 at 5:54

11 votes

Accepted

If a sequence grows too fast, then its harmonic sum cannot be rational

real-analysis sequences-and-series irrational-numbers rational-numbers

answered Mar 5, 2017 at 4:44

11 votes

Accepted

Proving that the iterated limit and the two dimensional limit are same

limits multivariable-calculus

answered Feb 1, 2015 at 17:47

11 votes

Accepted

How to deal with an indefinite L'Hôpital operation

calculus limits

answered Dec 1, 2015 at 4:21

11 votes

Showing that $\sup\{|f(x)-f(y)|, x,y\in X\}= \sup \ f - \inf \ f$

supremum-and-infimum

answered Apr 27, 2016 at 22:17

11 votes

Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$

integration measure-theory lebesgue-integral

answered Apr 11, 2016 at 13:46

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2,341 Answers

The math behind Warren Buffett's famous rule – never lose money

Prove $\lim_{n \to \infty} \frac{\ln(n)}{n}=0$ without L'Hospital's Rule

Integrability of Thomae's Function on $[0,1]$.

A nonnegative random variable has zero expectation if and only if it is zero almost surely

Why do some mathematical ideas seem counter-intuitive?

Proving that $\gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

Proof that if $f$ is integrable then also $f^2$ is integrable

Geometric mean of reals between 0 and 1

How to compute the pade approximation?

Uncorrelated but not independent random variables

How to evaluate the sum $\sum_{k=2}^{\infty}\log{(1-1/k^2)}$?

How to show that $\lim_{n\to \infty} \frac{a_1 +a_2 + \cdots + a_n}{n} = 0?$

Convergence $ \sum \frac{\sin \sqrt{n}}{n^{3/2}}, \quad \sum \frac{\sin n}{\sqrt{n}}, \quad\sum \frac{\sin \sqrt{n}}{n^{3/4}} $

Prove that $\lim\limits_{n\rightarrow\infty} \left(1+\frac{1}{a_{n}} \right)^{a_{n}}=e$ if $\lim\limits_{n\rightarrow\infty} a_{n}=\infty$

Numerical Integration for integrable singularity

On the convergence of $\sum_{n = 1}^\infty\frac{\sin\left(n^a\right)}{n^b}$

Can someone explain in general what a central difference formula is and what it is used for?

Proof of connection between improper Riemann Integral and Lebesgue integral

Evaluate the limit $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}\right)$

Evaluating a Lebesgue Integral

Deriving Taylor series without applying Taylor's theorem.

Riemann-Stieltjes integral of unbounded function

Is $\sum_{n=2}^{\infty}\frac{(-1)^n}{\sqrt n+(-1)^n}$ convergent?

Proof every convex function is continuous (Problem 10 Convex Functions Spivak)

Riemann integral on a single point

If a sequence grows too fast, then its harmonic sum cannot be rational

Proving that the iterated limit and the two dimensional limit are same

How to deal with an indefinite L'Hôpital operation

Showing that $\sup\{|f(x)-f(y)|, x,y\in X\}= \sup \ f - \inf \ f$

Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$