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Samrat Mukhopadhyay's user avatar
Samrat Mukhopadhyay's user avatar
Samrat Mukhopadhyay's user avatar
Samrat Mukhopadhyay
  • Member for 11 years
  • Last seen this week
56 votes

How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$?

25 votes
Accepted

$\sum a_n$ converges $\implies\ \sum a_n^2$ converges?

19 votes
Accepted

How to express the Frobenius norm of a matrix as the squared norm of its singular values?

18 votes

Prove that $ n < 2^{n}$ for all natural numbers $n$.

18 votes

How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?

17 votes

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

17 votes

How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$?

17 votes
Accepted

What functions satisfy such equation?

14 votes

Prove that $z_{n+1} = \frac12 \left(z_n +\frac{1}{z_n} \right) $ converges to $1$

11 votes
Accepted

Prove that $\sum_{r=1}^n \frac 1{r}\binom{n}{r} = \sum_{r=1}^n \frac 1{r}(2^r - 1)$

11 votes
Accepted

Show that a limit of a sequence is zero

10 votes

uniform random point in triangle in 3D

10 votes
Accepted

Find the value of $\,\, \lim_{n \to \infty}\Big(\!\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\!\Big)^{\!1/n} $

9 votes

Find the average of $\sin^{100} (x)$ in 5 minutes?

9 votes

Sum function of a series

8 votes
Accepted

complex series exponential evaluation

8 votes
Accepted

Integral $\int_{0}^{n^{2}} \lfloor \sqrt{t} \rfloor \rm dt $

8 votes

How to find the sum $\sum\limits_{k=1}^n (k^2+k+1)k!$?

8 votes

Integration by differentiating under the integral sign $I = \int_0^1 \frac{\arctan x}{x+1} dx$

8 votes

Find all roots of the equation $1-\frac{x}{1}+\frac{x(x-1)}{2!}-\cdots+(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0$

8 votes

How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$

7 votes
Accepted

Gaps between primes

7 votes

Prove that $\sqrt 5$ is irrational

7 votes

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

7 votes
Accepted

Evaluate $\lim_{n\to\infty} \frac{b_n}{a_n}$

6 votes
Accepted

Prove $\langle y,x \rangle \langle x,y \rangle \leq \langle y,y \rangle.$

6 votes
Accepted

If $|a+b|≤1,$ then $ |a|≤|b|+1.$

6 votes
Accepted

Statement $\frac ab+\frac bc+\frac ca \ge 3$ is true for any positive $a$, $b$, $c$?

6 votes

To find maximum possible value of this integral

6 votes

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

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