$\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9?$

score 294 · Answer 13 · 2026-03-29 04:01:46

294

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Fractions and percentages

Published on 29 Mar 2026 - 4:01

score 69 · Answer 14 · 2023-08-02 17:56:46

69

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define $f:(0,1]\to(0,1)$ by: if $x=\frac{1}{n}$ for $n\in\mathbb{N}$, then $f(x)=\frac{1}{n+1}$, else $f(x)=x$. Is this a bijection?

Published on 02 Aug 2023 - 17:56

score 59 · Answer 15 · 2026-03-26 04:30:16

59

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Jensen inequality with fraction expectation, determinant, and log function

Published on 26 Mar 2026 - 4:30

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$\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9?$

Fractions and percentages

define $f:(0,1]\to(0,1)$ by: if $x=\frac{1}{n}$ for $n\in\mathbb{N}$, then $f(x)=\frac{1}{n+1}$, else $f(x)=x$. Is this a bijection?

Jensen inequality with fraction expectation, determinant, and log function

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