s=(-1/2)1.(-1/2)2.(-1/2)3....(-1/2)2013.(-1/2)2014. Hỏi s âm hay dương
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\(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}}\)
Xét mẫu:
\(\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}\)
= \(\left(1+\frac{2013}{2}\right)+\left(1+\frac{2012}{3}\right)+...+\left(1+\frac{1}{2014}\right)+1\)
= \(\frac{2014}{2}+\frac{2014}{3}+....+\frac{2014}{2013}+\frac{2014}{2014}\)
= \(2014\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)\)
\(\Rightarrow S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{2014.\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)}\)
\(\Rightarrow S=\frac{1}{2014}\)
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S = 1 - 2 + 22 - 23+.....+ 22012 - 22013
2\(\times\)S = 2 - 22 + 23-.......- 22012 + 22013 - 22014
2 \(\times\) S + S = 1 - 22014
3S = 1 - 22014
3S - 22014 = 1 - 22014 - 22014 = 1 - 2.22014 = 1- 22015
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g: \(B=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot...\cdot\dfrac{19}{20}=\dfrac{1}{20}\)
h: \(=\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot..\cdot\dfrac{100}{99}=\dfrac{100}{2}=50\)
f: \(A=1+\dfrac{1}{2^{2014}}\)
\(B=\dfrac{2^{2014}+1+1}{2^{2014}+1}=1+\dfrac{1}{2^{2014}+1}\)
mà \(2^{2014}< 2^{2014}+1\)
nên A>B
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Ta có: \(\frac{1}{2^2}=\frac{1}{2.2}< \frac{1}{1.2}\)
Tương tự : \(\frac{1}{3^2}< \frac{1}{2.3}\); \(\frac{1}{4^2}< \frac{1}{3.4}\); ......... ; \(\frac{1}{2014^2}< \frac{1}{2013.2014}\)
\(\Rightarrow S< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+........+\frac{1}{2013.2014}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{2013}-\frac{1}{2014}\)
\(=1-\frac{1}{2014}=\frac{2013}{2014}\)
\(\Rightarrow S< \frac{2013}{2014}\left(đpcm\right)\)
Ta có:
\(S=\frac{-1}{2^1}\cdot\frac{1}{2^2}\cdot\frac{-1}{2^3}\cdot....\cdot\frac{-1}{2^{2013}}\cdot\frac{1}{2^{2014}}\)
\(S=\left(\frac{1}{2^2}\cdot\frac{1}{2^4}\cdot\frac{1}{2^6}\cdot...\cdot\frac{1}{2^{2014}}\right)\cdot\left(\frac{-1}{2}\cdot\frac{-1}{2^3}\cdot\frac{-1}{2^5}\cdot...\cdot\frac{-1}{2^{2013}}\right)\)
Gọi \(A=\frac{-1}{2}\cdot\frac{-1}{2^3}\cdot\frac{-1}{2^5}\cdot...\cdot\frac{-1}{2^{2013}}\)
Số phân số mang tử là -1 là
\(\frac{\left(2013-1\right)}{2}+1=1006+1=1007\)
Suy ra: \(A< 0\)
Mặt khác: \(\frac{1}{2^2}\cdot\frac{1}{2^4}\cdot\frac{1}{2^6}\cdot...\cdot\frac{1}{2^{2014}}>0\)
Do đó S là số âm