What is the tight upper bound of $ \sum_{i=1}^n \frac{i}{i+x_i}$ , where the $ x_i$ ‘s are distinct integers in $ \{1,2,…,n\}$ ?

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# Tag: $\sum_{i=1}^n

## Find the tight upper bound of $\sum_{i=1}^n \frac{i}{i+x_i}$, where the $x_i$’s are distinct in $\{1,2,…,n\}$

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What is the tight upper bound of $ \sum_{i=1}^n \frac{i}{i+x_i}$ , where the $ x_i$ ‘s are distinct integers in $ \{1,2,…,n\}$ ?