Question

So sánh:

S=\(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2013}{2^{2013}}\)  với 2

Answer 1

\(2S=1+\frac{2}{2}+\frac{3}{2^2}+........+\frac{2013}{2^{2012}}\)

\(2S-S=1+\frac{1}{2}+\frac{1}{2^2}+......+\frac{1}{2^{2012}}-\frac{2013}{2^{2013}}\)

\(S=1+\frac{1}{2}+.......+\frac{1}{2^{2012}}-\frac{2013}{2^{2013}}\)

\(S< 1+\frac{1}{2}+......+\frac{1}{2^{2012}}\)

\(2S< 2+1+.......+\frac{1}{2^{2011}}\)

\(2S-S< 2-\frac{1}{2^{2012}}\)

\(\Rightarrow S< 2-\frac{1}{2^{2012}}< 2\)

\(\Rightarrowđpcm\)

Answer 2

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