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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

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Function: parseLatexImg

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: model

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Function: require_once

수학동아 - 폴리매스
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[수학 정보] 특수한 합의 정수부분 구하기!
유클리드 2020.08.14 21:59 조회 782
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{100}}의 정수부분은 어떻게 계산할 수 있을까?

이 식만 본다면 이해가 힘들지만, 다음 식을 생각하면 어떨까?

\frac{1}{\sqrt{n}}=\frac{2}{2\sqrt{n}}=\frac{2}{\sqrt{n}+\sqrt{n}

여전히 파악을 못하겠는가? 힌트는 분모의 유리화와 부등식이다.

\frac{2}{\sqrt{n-1}+\sqrt{n}}<\frac{2}{\sqrt{n}+\sqrt{n}}<\frac{2}{\sqrt{n}+\sqrt{n+1}}

이제 눈치챘을 것이다. 여기서 분모의 유리화를 시도하자.

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

 이제 다 끝났다. 위 식의 정수부분은 18보다 크고, 대략 18.2보다 작으니, 정수부분은 18이다.

(1의 경우는 등호가 성립하지만, 나머지는 등호가 성립되지 않으니까 1의 경우는 무시하자)

 

 

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