Bài tập 16 trang 112 SGK Toán 10 NC
Bài tập 16 trang 112 SGK Toán 10 NC
Chứng minh rằng với mọi số nguyên n, ta có:
a) \(\frac{1}{{1.2}} + \frac{1}{{2.3}} + ... + \frac{1}{{n\left( {n + 1} \right)}} < 1\)
b) \(\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + ... + \frac{1}{{{n^2}}} < 2\)
a) Ta có \(\frac{1}{{k\left( {k + 1} \right)}} = \frac{1}{k} - \frac{1}{{k + 1}},\forall k \ge 1\)
Do đó:
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ... + \frac{1}{{n\left( {n + 1} \right)}}\\
= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + ... + \frac{1}{n} - \frac{1}{{n + 1}}
\end{array}\\
{ = 1 - \frac{1}{{n + 1}} < 1}
\end{array}\)
b) Ta có \(\frac{1}{{{k^2}}} < \frac{1}{{k\left( {k + 1} \right)}} \)
\(\Rightarrow \frac{1}{{{k^2}}} < \frac{1}{{k - 1}} - \frac{1}{k}\left( {k \ge 2} \right)\)
Do đó:
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + ... + \frac{1}{{{n^2}}}\\
< 1 + \left( {1 - \frac{1}{2} + ... + \frac{1}{{n - 1}} - \frac{1}{n}} \right)
\end{array}\\
\begin{array}{l}
\Rightarrow \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + ... + \frac{1}{{{n^2}}}\\
< 2 - \frac{1}{n} < 2
\end{array}
\end{array}\)
-- Mod Toán 10
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