So Sánh
S=\(\frac{1}{2}\)+\(\frac{2}{2^2}\)+\(\frac{3}{2^3}\)+...+\(\frac{2013}{2^{2013}}\)với 2
Những câu hỏi liên quan
So sánh:
S=\(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2013}{2^{2013}}\) với 2
\(2S=1+\frac{2}{2}+\frac{3}{2^2}+........+\frac{2013}{2^{2012}}\)
\(2S-S=1+\frac{1}{2}+\frac{1}{2^2}+......+\frac{1}{2^{2012}}-\frac{2013}{2^{2013}}\)
\(S=1+\frac{1}{2}+.......+\frac{1}{2^{2012}}-\frac{2013}{2^{2013}}\)
\(S< 1+\frac{1}{2}+......+\frac{1}{2^{2012}}\)
\(2S< 2+1+.......+\frac{1}{2^{2011}}\)
\(2S-S< 2-\frac{1}{2^{2012}}\)
\(\Rightarrow S< 2-\frac{1}{2^{2012}}< 2\)
\(\Rightarrowđpcm\)
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31 tháng 12 2017 lúc 22:12
Tính tổng :
\(S=\frac{1}{2013-1}+\frac{2}{2013+1}+\frac{2^2}{2013^2+1}+\frac{2^3}{2013^{2^2}+1}+.....+\frac{2^{n+1}}{2013^{2^n}+1}\)
So sánh P và Q biết : P = 2010/2011 + 2011/2012 + 2012/2013 và Q = 2010+2011+2012/ 2011 +2012+2013
Chứng tỏ N < 1 với N = \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}+\frac{1}{2010^2}\)
Ta có: \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2010^2}
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cho D = \(\frac{2^3}{3^3}+\frac{2^3}{5^3}+\frac{2^3}{7^3}+...+\frac{2^3}{2013^3}\).. So sánh D với 2/3
So sánh \(P=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....+\frac{1}{2013^2}+\frac{1}{2014^2}\)và \(Q=1\frac{3}{4}\)
cho so A=\(\frac{2013+\frac{1}{2}}{\left(2012+\frac{1}{2}\right)^2+2013+\frac{1}{2}}\)
B=\(\frac{2013+\frac{1}{3}}{\left(2012+\frac{1}{3}\right)^2+2013+\frac{1}{3}}\)
so sanh A va B
Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .
\(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}}\)
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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Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .
Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .