Problem(KAIST POW 2012-1)
Compute
(See KAIST POW 2012-1)
My solution
Let
By alternating series test(see Mathworld), and converge when .
Now let’s observe that
This gives us
Now we use the following well-known formula about series expansion of hyperbolic secant function(see Wolframfunction);
(it is sech(z), not sec(hz)…why LaTeX doesn’t have \sech operator command?? -_-)
Substitute , then we can get a closed expression of ;
Hence
Substitute , then we finally get
Hence, the answer is
Remark
We can see some of others’ solutions at the KAIST POW homepage (KAIST POW 2012-1). Most of the solutions I can see contain some algebraical identities about trigonometric functions. I think that there may exist a proof which uses telescoping; find such that
If exists, then the summation may become
I tried to find some properties of …But it was hard for me;;;
I can’t know there is a proof with telescoping method as I can’t see all of others’ solutions… but I believe that telescoping method may works if we find .
\sech가 안 먹히면 \operatorname{sech}를 써 주면 됩니다.
답변 감사합니다 ^^ {\rm sech}도 먹히는 듯 하군요. 근데 \rm과 \operatorname의 차이가 무엇인가요? \rm을 자주 사용했지만 커맨드가 정확히 어떤 짓을(?) 하는 건지 잘 모르겠네요.
\rm (혹은 \mathrm)은 로만 체를 입력할 때 씁니다. \operatorname{}은 함수를 쓸 때 주로 쓰는 걸로 보입니다. 적어도 위키피디아 상에선 표시상 큰 차이가 나진 않는 걸로 압니다.
그렇군요. LaTeX 매뉴얼을 좀 더 살펴볼 생각입니다. 좋은 정보 감사합니다!