Question

$\left(\sum_i a_i\right)^2\ge (n-1)\sum_i a_i^2 + b\implies 2a_i a_j \ge b/(n-1) \quad (\forall i\ne j)$

score 60 · Answer 1 · 2026-04-13 16:11:35

60

Views

$\left(\sum_i a_i\right)^2\ge (n-1)\sum_i a_i^2 + b\implies 2a_i a_j \ge b/(n-1) \quad (\forall i\ne j)$

Published on 13 Apr 2026 - 16:11

score 242 · Answer 2 · 2026-03-25 22:06:34

242

Views

Prove that the cyclic sum over $a, b, c > 0$ $ \sum (\frac{a}{a+b})^3 >= \frac{3}{8}$

Published on 25 Mar 2026 - 22:06

score 2.1k · Answer 3 · 2026-04-16 10:43:05

2.1k

Views

Sum of $s_n=10-8+6.4-5.12+...$

Published on 16 Apr 2026 - 10:43

score 1.4k · Answer 4 · 2026-04-17 06:20:01

1.4k

Views

Evaluate integral: $\int_0^{\frac{\pi}{2}}\ln(a^2\cos^2 x +b^2\sin^2x)dx$?

Published on 17 Apr 2026 - 6:20

score 75 · Answer 5 · 2026-04-17 09:49:15

75

Views

Evaluate $\frac{1}{1 \cdot 2 \cdot 3}+\frac{2}{4 \cdot 5 \cdot 6}+\frac{3}{7 \cdot 8 \cdot 9}+ \cdots$

Published on 17 Apr 2026 - 9:49

score 358 · Answer 6 · 2026-04-16 06:22:05

358

Views

For what value of $m$ the is sum $\sum_{i = 0}^{m} {10 \choose i}{20 \choose m - i}$ where ${p\choose q} = 0$, if $p<q$, a maximum

Published on 16 Apr 2026 - 6:22

score 112 · Answer 7 · 2026-04-11 06:40:30

112

Views

Summation coincides with integration?

Published on 11 Apr 2026 - 6:40

score 63 · Answer 8 · 2026-04-15 05:25:38

63

Views

For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$

Published on 15 Apr 2026 - 5:25

score 68 · Answer 9 · 2026-04-16 04:31:39

68

Views

Prove $2\sum_{k=0}^{\infty} \frac{(-1)^k(2k+1)}{(2k+1)^2 + 1} = \frac{\pi}{2}\mathrm{sech}(\frac{\pi}{2})$

Published on 16 Apr 2026 - 4:31

score 58 · Answer 10 · 2026-04-13 06:11:34

58

Views

Matrix norm proving calculation problem.

Published on 13 Apr 2026 - 6:11

score 50 · Answer 11 · 2026-04-14 11:38:48

50

Views

How to proceed with the differential equation $x^{\left(x\right)}=0$, where $x\in\mathbb{N}$?

Published on 14 Apr 2026 - 11:38

score 740 · Answer 12 · 2026-04-15 18:13:32

Consider the function $f (z) = z +2z^2 +3z^3 +··· = \sum_{n≥0} nz^n$ defined on the open disk $\{z \vert|z| < 1\}$. Choose the correct option

score 100 · Answer 13 · 2026-03-27 01:12:58

100

Views

What's this partial sum $ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

Published on 27 Mar 2026 - 1:12

score 199 · Answer 14 · 2026-03-25 19:04:33

199

Views

A summation similar to Vandermonde identity

Published on 25 Mar 2026 - 19:04

score 189 · Answer 15 · 2026-04-09 04:59:31

189

Views

How do I simplify this sum of arccosines?

Published on 09 Apr 2026 - 4:59

List Question

$\left(\sum_i a_i\right)^2\ge (n-1)\sum_i a_i^2 + b\implies 2a_i a_j \ge b/(n-1) \quad (\forall i\ne j)$

Prove that the cyclic sum over $a, b, c > 0$ $ \sum (\frac{a}{a+b})^3 >= \frac{3}{8}$

Sum of $s_n=10-8+6.4-5.12+...$

Evaluate integral: $\int_0^{\frac{\pi}{2}}\ln(a^2\cos^2 x +b^2\sin^2x)dx$?

Evaluate $\frac{1}{1 \cdot 2 \cdot 3}+\frac{2}{4 \cdot 5 \cdot 6}+\frac{3}{7 \cdot 8 \cdot 9}+ \cdots$

For what value of $m$ the is sum $\sum_{i = 0}^{m} {10 \choose i}{20 \choose m - i}$ where ${p\choose q} = 0$, if $p<q$, a maximum

Summation coincides with integration?

For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$

Prove $2\sum_{k=0}^{\infty} \frac{(-1)^k(2k+1)}{(2k+1)^2 + 1} = \frac{\pi}{2}\mathrm{sech}(\frac{\pi}{2})$

Matrix norm proving calculation problem.

How to proceed with the differential equation $x^{\left(x\right)}=0$, where $x\in\mathbb{N}$?

Consider the function $f (z) = z +2z^2 +3z^3 +··· = \sum_{n≥0} nz^n$ defined on the open disk $\{z \vert|z| < 1\}$. Choose the correct option

What's this partial sum $ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

A summation similar to Vandermonde identity

How do I simplify this sum of arccosines?

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