User Paramanand Singh

81 votes

3 answers

5k views

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

asked Jul 19, 2014 at 10:37

53 votes

14 answers

5k views

Simplest way to get the lower bound $\pi > 3.14$

real-analysis sequences-and-series inequality pi upper-lower-bounds

asked Oct 23, 2017 at 6:45

45 votes

2 answers

3k views

An infinite series plus a continued fraction by Ramanujan

sequences-and-series continued-fractions

asked Jun 14, 2014 at 13:53

42 votes

2 answers

7k views

What is the use of hyperreal numbers?

analysis epsilon-delta applications nonstandard-analysis infinitesimals

asked Jun 23, 2016 at 7:49

32 votes

7 answers

17k views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

calculus limits alternative-proof

asked Jul 7, 2013 at 5:34

25 votes

4 answers

3k views

Proof of a Ramanujan Integral

calculus definite-integrals improper-integrals

asked May 25, 2014 at 5:10

25 votes

4 answers

3k views

Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$

sequences-and-series

asked Aug 24, 2014 at 7:57

25 votes

1 answer

2k views

Gosper's unusual formula connecting $e$ and $\pi$

sequences-and-series pi

asked Feb 1, 2015 at 6:27

25 votes

2 answers

9k views

Taylor's Theorem with Peano's Form of Remainder

calculus

asked Jun 2, 2016 at 5:37

23 votes

1 answer

525 views

A problem posed by Ramanujan involving $\sum e^{-5\pi n^2}$

sequences-and-series theta-functions

asked Jun 5, 2020 at 2:19

22 votes

1 answer

783 views

Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

sequences-and-series

asked Aug 16, 2014 at 7:56

19 votes

1 answer

2k views

An alternative proof of Cauchy's Mean Value Theorem

calculus real-analysis

asked Aug 1, 2015 at 5:17

19 votes

1 answer

607 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

sequences-and-series

asked Jun 1, 2014 at 10:11

19 votes

6 answers

3k views

For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.

calculus real-analysis

asked Feb 15, 2014 at 5:49

19 votes

3 answers

3k views

Monotone functions and non-vanishing of derivative

calculus real-analysis

asked Jul 1, 2016 at 15:07

16 votes

2 answers

431 views

If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$, then show that $f(x) \to A$ as $x \to \infty$

calculus real-analysis ordinary-differential-equations limits convergence-divergence

asked Mar 25, 2014 at 11:24

15 votes

2 answers

2k views

Doubt regarding proof that $\mathbb{Z}, \mathbb{Z}[x]$ are unique factorization domains

abstract-algebra ring-theory unique-factorization-domains

asked Apr 15, 2017 at 6:59

15 votes

2 answers

465 views

Ramanujan's series $1+\sum_{n=1}^{\infty}(8n+1)\left(\frac{1\cdot 5\cdots (4n-3)}{4\cdot 8\cdots (4n)}\right)^{4}$

sequences-and-series

asked Nov 4, 2017 at 8:50

14 votes

3 answers

2k views

Continuous decreasing function has a fixed point

calculus real-analysis

asked Mar 23, 2014 at 10:21

14 votes

1 answer

570 views

Continued fraction estimation of error in Leibniz series for $\pi$.

calculus reference-request pi continued-fractions

asked Jul 19, 2015 at 7:59

14 votes

2 answers

698 views

Bellard's exotic formula for $\pi$

soft-question pi

asked Jun 13, 2015 at 6:28

14 votes

1 answer

791 views

Unsolvability of a Quintic and its link with "Simplicity" of $A_{5}$

abstract-algebra galois-theory

asked Jun 28, 2015 at 5:54

14 votes

3 answers

1k views

Show that $\sum\limits_n1/x_{n}^{2} = 1/10$ where $x_{n}$ is the $n^{\text{th}}$ positive root of $\tan x = x$ [duplicate]

sequences-and-series

asked Jun 29, 2014 at 5:32

13 votes

4 answers

4k views

Continued fraction for $\tan(nx)$

calculus algebra-precalculus trigonometry continued-fractions

asked Jun 30, 2013 at 7:02

13 votes

0 answers

229 views

Bi-linear relation between two continued fractions

continued-fractions

asked Aug 10, 2015 at 11:48

13 votes

2 answers

446 views

Limit of $\sin (a^{n}\theta\pi)$ as $ n \to \infty$ where $a$ is an integer greater than $2$

calculus

asked Jul 20, 2013 at 14:00

12 votes

1 answer

615 views

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

calculus integration definite-integrals

asked Jul 9, 2015 at 6:00

11 votes

1 answer

441 views

Ramanujan's transformation formula connected with $r_{2}(n)$

sequences-and-series number-theory reference-request

asked Jul 11, 2015 at 8:56

11 votes

1 answer

238 views

If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$

calculus real-analysis

asked Mar 18, 2014 at 17:57

11 votes

1 answer

572 views

Evaluation of complete elliptic integrals $K(k) $ for $k=\tan(\pi/8),\sin(\pi/12)$

calculus integration definite-integrals elliptic-integrals

asked Aug 12, 2017 at 12:04

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120 Questions

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

Simplest way to get the lower bound $\pi > 3.14$

An infinite series plus a continued fraction by Ramanujan

What is the use of hyperreal numbers?

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

Proof of a Ramanujan Integral

Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$

Gosper's unusual formula connecting $e$ and $\pi$

Taylor's Theorem with Peano's Form of Remainder

A problem posed by Ramanujan involving $\sum e^{-5\pi n^2}$

Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

An alternative proof of Cauchy's Mean Value Theorem

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.

Monotone functions and non-vanishing of derivative

If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$, then show that $f(x) \to A$ as $x \to \infty$

Doubt regarding proof that $\mathbb{Z}, \mathbb{Z}[x]$ are unique factorization domains

Ramanujan's series $1+\sum_{n=1}^{\infty}(8n+1)\left(\frac{1\cdot 5\cdots (4n-3)}{4\cdot 8\cdots (4n)}\right)^{4}$

Continuous decreasing function has a fixed point

Continued fraction estimation of error in Leibniz series for $\pi$.

Bellard's exotic formula for $\pi$

Unsolvability of a Quintic and its link with "Simplicity" of $A_{5}$

Show that $\sum\limits_n1/x_{n}^{2} = 1/10$ where $x_{n}$ is the $n^{\text{th}}$ positive root of $\tan x = x$ [duplicate]

Continued fraction for $\tan(nx)$

Bi-linear relation between two continued fractions

Limit of $\sin (a^{n}\theta\pi)$ as $ n \to \infty$ where $a$ is an integer greater than $2$

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

Ramanujan's transformation formula connected with $r_{2}(n)$

If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$

Evaluation of complete elliptic integrals $K(k) $ for $k=\tan(\pi/8),\sin(\pi/12)$