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Sequences and series › Finite sums › Alternating sums

Alternating sum of minimum values

High school · ●●●○○ 50 pts

For a positive integer nn, define fn(x)=(∑i=1n∣x−i∣)2−∑i=1n(x−i)2f_n(x)=\left(\sum_{i=1}^{n}|x-i|\right)^2-\sum_{i=1}^{n}(x-i)^2 on the interval [0,n+1][0,n+1]. Let ana_n be the minimum value of fn(x)f_n(x). Find ∑n=111(−1)n+1an\sum_{n=1}^{11}(-1)^{n+1}a_n.

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absolute value functionspiecewise functionssum of squares