Problem 2

Let \(A=\{1,2,\ldots,n\}\) and \(C\) is the set of all bijective function from \(A\) to itself. Define the function \(T:C\rightarrow \mathbb{R}\) as follow: for all \(f\in C\), \(T(f)\) be the number of pair \((x,y)\in A\times A\) such that \(f(x)>f(y)\) whenever \(x<y\). Evaluate the following summation!
\[\sum\limits_{f\in C}\prod\limits_{x=1}^n (-1)^{T(f)}x^{f(x)}\]

Problem 1

Let \(\{x_n\}\) be a sequence of positive real number that converges to 0. For all \(n\in \mathbb{N}\), define \[a_n=\left(\sum\limits_{k=1}^{n}x_k^n\right)^{\frac{1}{n}}.\] Prove that the sequence \(\{a_n\}\) converges and find its limit!

Welcome Mathster!

Hello mathster!
You've found the right place to treat yourself with many challenging and enjoyable math problems. Let me know your answer by submit it to garryarielcussoy@gmail.com. For you who becomes the first to solve it correctly (in the period time of a problem), i will post your answer. And of course, if you have some math problems too, you can submit it to me, and i will post it here. So, lets get started, and enjoy the sensation of doing math :)