Solusi IMC 2015 Hari 2 Soal 6

Perhatikan bahwa

\displaystyle \frac{1}{\sqrt{n}(n+1)} < \frac{2}{\sqrt{n}} - \frac{2}{\sqrt{n+1}}

\displaystyle \Leftrightarrow 1 \le 2\sqrt{n+1}(\sqrt{n+1}-\sqrt{n})

\displaystyle \Leftrightarrow \sqrt{n} < \sqrt{n+1}.

Dengan demikian, untuk setiap m \ge n, berlaku

\displaystyle \sum_{n=1}^m \frac{1}{\sqrt{n}(n+1)} < \sum_{n=1}^m \frac{2}{\sqrt{n}}-\frac{2}{\sqrt{n+1}} = \frac{2}{\sqrt{1}}-\frac{2}{\sqrt{m+1}}.

Ambil limit m \rightarrow \infty di kedua ruas, maka diperoleh

\displaystyle \lim_{m \rightarrow \infty} \sum_{n=1}^m \frac{1}{\sqrt{n}(n+1)} < 2.

Jadi, pernyataan pada soal terbukti.

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