User Paramanand Singh

66 votes

Why does simplifying a function give it another limit

answered Jun 12, 2016 at 1:14

43 votes

Accepted

Ramanujan's approximation for $\pi$

calculus power-series approximation pi

answered Sep 7, 2014 at 5:39

42 votes

Accepted

How to calculate $\lim_{x \to 0} \frac{x^{6000} - (\sin x)^{6000}}{x^2(\sin x)^{6000}}$?

calculus limits

answered May 15, 2016 at 16:38

37 votes

Accepted

The Main Theorems of Calculus

calculus real-analysis

answered May 16, 2016 at 8:05

37 votes

Accepted

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

abstract-algebra galois-theory

answered Oct 20, 2013 at 8:11

34 votes

Accepted

Finding a limit using change of variable- how come it works?

calculus limits

answered Dec 18, 2014 at 7:12

31 votes

Accepted

Motivation for/history of Jacobi's triple product identity

sequences-and-series math-history motivation q-series

answered Jul 12, 2013 at 8:57

30 votes

Accepted

Infinite Integration by Parts

calculus integration sequences-and-series

answered Feb 15, 2017 at 4:53

30 votes

Accepted

What even are elliptic functions?

complex-analysis special-functions analytic-number-theory elliptic-integrals elliptic-functions

answered Oct 10, 2019 at 15:19

28 votes

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

sequences-and-series complex-analysis integration summation contour-integration

answered May 25, 2016 at 4:01

27 votes

Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

real-analysis sequences-and-series summation

answered Oct 3, 2017 at 11:52

25 votes

Motivation for Ramanujan's mysterious $\pi$ formula

calculus sequences-and-series approximation pi

answered Jul 2, 2013 at 5:16

23 votes

Accepted

Why isn't $\int \frac{1}{x}~dx = \frac{x^0}{0}$?

calculus integration indefinite-integrals

answered Jan 30, 2017 at 8:34

22 votes

"Gaps" or "holes" in rational number system

real-analysis analysis rational-numbers

answered Apr 30, 2020 at 3:15

21 votes

Accepted

If $K$ be an algebraic extension of $E$ and $E$ be an algebraic extension of $F$ then $K$ is an algebraic extension of $F$.

abstract-algebra extension-field

answered Apr 20, 2017 at 3:52

21 votes

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

calculus real-analysis derivatives definition

answered Oct 27, 2013 at 10:41

20 votes

Accepted

Compute $\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx$

calculus integration limits definite-integrals

answered Jan 30, 2020 at 8:37

19 votes

Accepted

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

sequences-and-series reference-request theta-functions

answered May 21, 2016 at 3:50

19 votes

Accepted

Analyzing limits problem Calculus (tell me where I'm wrong).

calculus limits

answered May 13, 2016 at 14:34

18 votes

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

big-list irrational-numbers alternative-proof rationality-testing

answered Jun 14, 2015 at 5:03

18 votes

Proving $f'(1)$ exist for $f$ satisfying $f(xy)=xf(y)+yf(x)$

calculus real-analysis derivatives functional-equations

answered Jan 10, 2017 at 4:49

18 votes

Accepted

Taylor's Theorem with Peano's Form of Remainder

calculus

answered Jun 2, 2016 at 5:51

17 votes

Accepted

On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $

sequences-and-series closed-form gamma-function bernoulli-numbers

answered Sep 27, 2016 at 20:56

17 votes

Accepted

Limits and algebraic simplification

calculus limits infinity

answered Sep 23, 2017 at 15:27

17 votes

Is there a "positive" definition for irrational numbers?

irrational-numbers

answered Sep 6, 2016 at 11:35

17 votes

Accepted

Perfect understanding of Riemann Sums

calculus integration riemann-sum

answered Oct 27, 2018 at 6:43

15 votes

Accepted

Why is the upper Riemann integral the infimum of all upper sums?

real-analysis riemann-integration

answered Dec 7, 2016 at 12:28

14 votes

Accepted

Taking limits on each term in inequality invalid?

real-analysis limits inequality proof-verification

answered Jul 5, 2015 at 5:04

14 votes

Prove that $\sin n\theta=n\sin \theta-\frac{n(n^2-1)}{3!}\sin^3\theta+\frac{n(n^2-1)(n^2-3^2)}{5!}\sin^5\theta+\cdots$

calculus real-analysis sequences-and-series trigonometry reference-request

answered Apr 12, 2016 at 13:00

14 votes

When converting a Riemann sum to an integral, how does one decide the bounds of the converted Riemann sum?

calculus integration

answered May 20, 2020 at 9:09

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2,439 Answers

Why does simplifying a function give it another limit

Ramanujan's approximation for $\pi$

How to calculate $\lim_{x \to 0} \frac{x^{6000} - (\sin x)^{6000}}{x^2(\sin x)^{6000}}$?

The Main Theorems of Calculus

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

Finding a limit using change of variable- how come it works?

Motivation for/history of Jacobi's triple product identity

Infinite Integration by Parts

What even are elliptic functions?

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

Motivation for Ramanujan's mysterious $\pi$ formula

Why isn't $\int \frac{1}{x}~dx = \frac{x^0}{0}$?

"Gaps" or "holes" in rational number system

If $K$ be an algebraic extension of $E$ and $E$ be an algebraic extension of $F$ then $K$ is an algebraic extension of $F$.

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Compute $\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx$

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

Analyzing limits problem Calculus (tell me where I'm wrong).

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

Proving $f'(1)$ exist for $f$ satisfying $f(xy)=xf(y)+yf(x)$

Taylor's Theorem with Peano's Form of Remainder

On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $

Limits and algebraic simplification

Is there a "positive" definition for irrational numbers?

Perfect understanding of Riemann Sums

Why is the upper Riemann integral the infimum of all upper sums?

Taking limits on each term in inequality invalid?

Prove that $\sin n\theta=n\sin \theta-\frac{n(n^2-1)}{3!}\sin^3\theta+\frac{n(n^2-1)(n^2-3^2)}{5!}\sin^5\theta+\cdots$

When converting a Riemann sum to an integral, how does one decide the bounds of the converted Riemann sum?