Question

Cho S=1/5+2/5^2+3/5^3+...+2012/5^2012 Hãy so sánh S với 1/3

Answer 1

Lời giải:

\(S=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...+\frac{2012}{5^{2012}}\)

\(\Rightarrow 5S=1+\frac{2}{5}+\frac{3}{5^2}+...+\frac{2012}{5^{2011}}\)

Trừ theo vế:
\(4S=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2011}}-\frac{2012}{5^{2012}}\)

\(20S=5+1+\frac{1}{5}+...+\frac{1}{5^{2010}}-\frac{2012}{5^{2011}}\)

Trừ theo vế:
\(16S=5-\frac{2012}{5^{2011}}-\frac{1}{5^{2011}}+\frac{2012}{5^{2012}}\)

\(16S=5-\frac{2013}{5^{2011}}+\frac{2012}{5^{2012}}< 5-\frac{2013}{5^{2011}}+\frac{2013}{5^{2011}}=5\)

\(S< \frac{5}{16}< \frac{1}{3}\)