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[창의 퍼즐] [정수] 가우스 기호+텔레스코핑

mwryan 2020.09.28 18:16 조회 690

문제 푼 사람 1명

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

$\sum_{k=1}^\infty\[\frac{a}{2^k}+\frac{1}{2}]$

다음 급수를 더하면 어떤 값에 수렴하는지 텔레스코핑(망원급수)을 사용하여 구하시오.(단, a는 자연수)

이전글 [질문]무한의 사칙연산

다음글 정육면체의 꼭짓점 지나기[도형]

이 문제 어떠셨나요?

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계산귀찮다 Lv.1 2021.01.10 07:57

에르미트항등식 쓰고 텔레스코핑하면 되겠네 계산은 밑에서 알아서 해주세요

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해결

도전

BlueBox Lv.2 2021.01.12 01:53

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1. 에르미트 항등식을 생각해보면 아래의 식을 생각할 수 있다.

   $[x]+[x+\frac{1}{2}]=[2x]$

2. 1의 식에서 x에   $\frac{a}{2^{k}}$ 를 대입하면 아래의 식이 된다.

$[\frac{a}{2^{k}}+\frac{1}{2}]=[\frac{a}{2^{k-1}}]-[\frac{a}{2^{k}}]$

3. 2의 식을 통해 텔레스코핑을 돌린다.

   $\sum_{k=1}^{\infty }[\frac{a}{2^{k}}+\frac{1}{2}]=\sum_{k=1}^{\infty }([\frac{a}{2^{k-1}}]-[\frac{a}{2^{k}}])=([\frac{a}{2^{0}}]-{\color{Red} [\frac{a}{2^{1}}]}) + ({\color{Red} [\frac{a}{2^{1}}]}-{\color{Red} [\frac{a}{2^{2}}]}) + ... + ({\color{Red} [\frac{a}{2^{\infty -1}}]}-{\color{Red} [\frac{a}{2^{\infty }}]})={\color{Blue} [a]}$

따라서 $\sum_{k=1}^{\infty }[\frac{a}{2^{k}}+\frac{1}{2}]=[a]$

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