昨日参加了北京大学的硕士入学考试，遇到了一道令人兴奋但难以解决的问题：证明两个级数等式之间的关系。该问题的核心在于如何处理左边分数中的n次幂，并将其转化为积分形式。这是一个对数学技能的极大挑战。
Yesterday I took part in the Masters Entrance Exam of Peking University. It was really inspiring! Here is a problem that I understand, have tried true, but do not know how to solve: Prove that ∑k=0m(−1)k(mk)1k+n+1=∑k=0n(−1)k(nk)1k+m+1,\sum_{k=0}^m(-1)^k{m\choose k}\frac{1}{k+n+1}=\sum_{k=0}^n(-1)^k{n\choose k}\frac{1}{k+m+1},k=0∑m(−1)k(km)k+n+11=k=0∑n(−1)k(kn)k+m+11, where mmm and nnn are positive integers.
Hint: The difficult part is to get rid of the nnn in the denominator on the left. Turn the fraction into an integral.
Challenge: Can you find a closed-form solution?
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