수학동아 - 폴리매스

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함께 풀고 싶은 문제

나도 수학쌤 문장제 문제를 변형해 문제를 내는 곳입니다.

[문장제 문제&개념응용] 수렴or 발산

sicma 2021.06.10 10:23 조회 409

문제 푼 사람 0명

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

모든 소수의 합은 무엇일까? 당연히 발산할 것이다.

그렇다면 모든 소수의 역수의 합은 수렴하는가 아니면 발산하는가?

또한 수렴한다면 수렴값을 알 수 있을까?

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

모든 소수의 곱은 무엇인가? 이것 역시 발산한다.

그렇다면 $a_{n}=\frac{p_{2n-1}}{p_{2n}}$ 과 같이 정의하자.

(p_k는 k 번째 소수이다. 즉 p_2n-1,p_2n은 각각 (2n-1)번째, 2n 반째 소수인 것이다.

a₁,a₂, 부터 쭉 곱해나갈때 수렴할까? 발산할까?

수렴한다면 수렴값을 알 수 있을까?

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

연속한 삼각수 n개의 합을 A_n이라 하자.

처음에 어떤 삼각수 부터 더했는지는 모르나 A_x는 변하지 않는 상수이다.

A₁,A₂,... 중에서 하나를 뽑았을 때 A_k가 뽑힐 것이다.

A_k의 약수의 갯수가 k를 나눌 확률은 얼마인가?

(A₁=1,3,6,10,15,...

A₂=4,9,16,25,...

A₃=10,19,31,46,...

이전글 일륜차(인디언들의 반란)

다음글 원과 정사각형

이 문제 어떠셨나요?

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피카파이 Lv.8 2021.09.02 07:17 비밀댓글

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