User Nolord

10 votes

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How can I solve to a closed form of the sequence $c_n=-\frac{1}{n}\sum_{k=0}^{n-1}c_{n-k-1}a_{k+1}$, with $c_0=1$?

answered Jun 22 at 20:13

8 votes

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I need help to evaluate Integral :$I=\int_0^{\pi} \frac{\arctan^2(\sin(x))}{1+\cos(x)}dx$

answered Jun 23 at 18:04

6 votes

Finding the smallest perfect square whose last 3 digits are the same. How do I do this by hand?

answered Jul 30 at 0:40

6 votes

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Limiting growth ratio of $[x^n]f(x)^n$

answered May 29 at 9:28

4 votes

Counting ones in binary representation: When is the product multiplicative?

answered Jun 8 at 17:05

4 votes

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The umbral calculus proof of the higher order product rule

answered Jan 7 at 15:19

3 votes

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Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

answered Jul 7 at 4:20

3 votes

Approximating $\log x$ by a sum of power functions $a x^b$

answered Jun 8 at 8:48

2 votes

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Overkill definition of finite and infinte sums in $\mathbb{R}$

answered Aug 26 at 12:47

2 votes

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Showing $\sum_{k=0}^{n} 2^{2k}\binom{n}{k}\binom{k}{n-k}=2^{n}\binom{2n}{n}$, and another identity, combinatorially

answered Aug 28 at 18:29

2 votes

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Is there an elegant method to find whether the graph is symmetrical or not

answered Jun 29 at 8:15

2 votes

Maximum number of black cells in a table where each cell has at most two adjacent black cells: Generalizations of JBMO 2019 P4

answered Aug 15 at 12:13

2 votes

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Sequences of the form $A(n) = A(A(n-1)\bmod n)^2$

answered Apr 22 at 15:08

2 votes

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High school inequality: how to prove that $\lim_{n \to \infty} \left( \prod_{r = 1} ^ n \frac{3 ^ r}{3 ^ r - 1}\right) < 2?$

answered Apr 19 at 12:32

2 votes

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Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?

answered Apr 19 at 16:17

1 vote

Ramanujan's series for the exponential integral

answered Apr 19 at 10:39

1 vote

How to give a closed form to $e^{a(x) \frac{d}{dx} + b(x)I}[f]$ in physicists style abuse of notation?

answered May 26 at 7:28

1 vote

When exactly are two functions said to be equal?

answered Jun 23 at 7:31

1 vote

How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$

answered Apr 23 at 15:02

1 vote

Multinomial theorem for matrices

answered Jul 30 at 1:13

1 vote

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Weighted sums of partial Bell polynomials

answered Aug 2 at 23:33

0 votes

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How to prove that the sequence $a_n=\lfloor r \rfloor + \frac{\lfloor (r- \lfloor r \rfloor) \cdot 10^n \rfloor}{10^n}$ converges to $r$?

answered Aug 17 at 18:20

0 votes

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Why does $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} = L \implies f = \Theta(g)$ not hold when $L=0$?

answered Jul 7 at 8:21

0 votes

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Notation for non-commutative $(A+B)^n$ in developped form

answered Jul 31, 2023 at 6:33

0 votes

Evaluate Limit of Faulhaber Formula

answered Jun 6 at 7:00

0 votes

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Proof of a Limit related to Gauss' Convergence test

answered Jun 9 at 7:19

0 votes

Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?

answered Jan 8 at 16:10

0 votes

Expressing $\frac{2}{n}$ as the sum of two unit fractions

answered Apr 21 at 8:01

-1 votes

Irreducibility of $x^8+6x^4+1$

answered Aug 29 at 9:04

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29 Answers

How can I solve to a closed form of the sequence $c_n=-\frac{1}{n}\sum_{k=0}^{n-1}c_{n-k-1}a_{k+1}$, with $c_0=1$?

I need help to evaluate Integral :$I=\int_0^{\pi} \frac{\arctan^2(\sin(x))}{1+\cos(x)}dx$

Finding the smallest perfect square whose last 3 digits are the same. How do I do this by hand?

Limiting growth ratio of $[x^n]f(x)^n$

Counting ones in binary representation: When is the product multiplicative?

The umbral calculus proof of the higher order product rule

Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

Approximating $\log x$ by a sum of power functions $a x^b$

Overkill definition of finite and infinte sums in $\mathbb{R}$

Showing $\sum_{k=0}^{n} 2^{2k}\binom{n}{k}\binom{k}{n-k}=2^{n}\binom{2n}{n}$, and another identity, combinatorially

Is there an elegant method to find whether the graph is symmetrical or not

Maximum number of black cells in a table where each cell has at most two adjacent black cells: Generalizations of JBMO 2019 P4

Sequences of the form $A(n) = A(A(n-1)\bmod n)^2$

High school inequality: how to prove that $\lim_{n \to \infty} \left( \prod_{r = 1} ^ n \frac{3 ^ r}{3 ^ r - 1}\right) < 2?$

Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?

Ramanujan's series for the exponential integral

How to give a closed form to $e^{a(x) \frac{d}{dx} + b(x)I}[f]$ in physicists style abuse of notation?

When exactly are two functions said to be equal?

How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$

Multinomial theorem for matrices

Weighted sums of partial Bell polynomials

How to prove that the sequence $a_n=\lfloor r \rfloor + \frac{\lfloor (r- \lfloor r \rfloor) \cdot 10^n \rfloor}{10^n}$ converges to $r$?

Why does $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} = L \implies f = \Theta(g)$ not hold when $L=0$?

Notation for non-commutative $(A+B)^n$ in developped form

Evaluate Limit of Faulhaber Formula

Proof of a Limit related to Gauss' Convergence test

Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?

Expressing $\frac{2}{n}$ as the sum of two unit fractions

Irreducibility of $x^8+6x^4+1$