Question

Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

score 523 · Answer 1 · 2026-04-09 10:08:05

523

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Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Published on 09 Apr 2026 - 10:08

score 81 · Answer 2 · 2026-05-11 07:40:47

81

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Contradiction on an Inequality

Published on 11 May 2026 - 7:40

score 80 · Answer 3 · 2026-04-16 00:14:02

80

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Evaluate the following series $ \sum_{n=0}^{\infty} \big(e^{(4n+1)\pi\sqrt{3}}+2+e^{-(4n+1)\pi\sqrt{3}}\big)^{-1} $

Published on 16 Apr 2026 - 0:14

score 372 · Answer 4 · 2026-04-14 16:17:36

372

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$ \sum_{n=1}^\infty \csc^2(\omega\pi n)= \frac{A}{\pi} +B $

Published on 14 Apr 2026 - 16:17

score 80 · Answer 5 · 2026-04-16 08:34:48

80

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Inequality related to $\sum_{i=1}^{m}\sum_{j=i+1}^{m} \left\Vert a_{i}+a_{j}\right\Vert$

Published on 16 Apr 2026 - 8:34

score 463 · Answer 6 · 2026-04-16 12:10:15

463

Views

Find $\sum_{r=1}^{3n-1}{ (-1)^{r-1}r\over{3n \choose r}}$, if $n$ is even

Published on 16 Apr 2026 - 12:10

score 74 · Answer 7 · 2026-04-09 11:23:17

74

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What Inequality holds true if and only if when $|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$ is true?

Published on 09 Apr 2026 - 11:23

score 51 · Answer 8 · 2026-05-15 04:33:38

51

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Obliterating Squares and Roots

Published on 15 May 2026 - 4:33

score 127 · Answer 9 · 2026-04-15 16:03:42

127

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Is it true that $\sum_{i=0}^{n-1} 2^i$ divides $\sum_{i=n}^{2n-1} 2^i$?

Published on 15 Apr 2026 - 16:03

score 44 · Answer 10 · 2026-04-17 10:27:22

44

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Show $ f(x)=\frac{x^2}{(1-x^2)^2}-\frac{3x^2}{(3-x^2)^2}, |x|<1 $ is equal to $ f(x)=\sum_{n=0}^\infty{n(1-3^{-n})x^{2n}}, |x|<R $

Published on 17 Apr 2026 - 10:27

score 101 · Answer 11 · 2026-04-14 03:15:56

101

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Help with proof that $ \sum_{i=0}^k{k \choose i}{N-k \choose n-i} = {N \choose n} $

Published on 14 Apr 2026 - 3:15

score 57 · Answer 12 · 2026-04-11 15:29:51

57

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Existence of two sequences

Published on 11 Apr 2026 - 15:29

score 45 · Answer 13 · 2026-04-10 01:54:30

45

Views

How can I prove that the sum of segments is infinite?

Published on 10 Apr 2026 - 1:54

score 91 · Answer 14 · 2026-04-15 18:08:07

91

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Prove that $\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor$.

Published on 15 Apr 2026 - 18:08

score 103 · Answer 15 · 2026-04-11 12:51:18

103

Views

Rewriting using sigma notation

Published on 11 Apr 2026 - 12:51

List Question

Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Contradiction on an Inequality

Evaluate the following series $ \sum_{n=0}^{\infty} \big(e^{(4n+1)\pi\sqrt{3}}+2+e^{-(4n+1)\pi\sqrt{3}}\big)^{-1} $

$ \sum_{n=1}^\infty \csc^2(\omega\pi n)= \frac{A}{\pi} +B $

Inequality related to $\sum_{i=1}^{m}\sum_{j=i+1}^{m} \left\Vert a_{i}+a_{j}\right\Vert$

Find $\sum_{r=1}^{3n-1}{ (-1)^{r-1}r\over{3n \choose r}}$, if $n$ is even

What Inequality holds true if and only if when $|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$ is true?

Obliterating Squares and Roots

Is it true that $\sum_{i=0}^{n-1} 2^i$ divides $\sum_{i=n}^{2n-1} 2^i$?

Show $ f(x)=\frac{x^2}{(1-x^2)^2}-\frac{3x^2}{(3-x^2)^2}, |x|<1 $ is equal to $ f(x)=\sum_{n=0}^\infty{n(1-3^{-n})x^{2n}}, |x|<R $

Help with proof that $ \sum_{i=0}^k{k \choose i}{N-k \choose n-i} = {N \choose n} $

Existence of two sequences

How can I prove that the sum of segments is infinite?

Prove that $\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor$.

Rewriting using sigma notation

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