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19. √[x/(x^2+x+1)]+√[x/(x^2-x+1)]
= √[1/(x+1+1/x)]+√[1/(x-1+1/x)]
= 1/√[(x+1/x)^2-1]+1/√[(x+1/x)^2-3]
= 1/2+1/√2 = (1+√2)/2
18 通项 1/[(n+1)√n+n√(n+1)]
= [(n+1)√n-n√(n+1)]/[n(n+1)^2-(n+1)n^2]
= [(n+1)√n-n√(n+1)]/[n(n+1)]
= 1/√n - 1/√(n+1)
则 原式 = 1-1/√2+1/√2-1/√3+...+1/√2024-1/√2025
= 1-1/√2025 = 1-1/45 = 44/45
= √[1/(x+1+1/x)]+√[1/(x-1+1/x)]
= 1/√[(x+1/x)^2-1]+1/√[(x+1/x)^2-3]
= 1/2+1/√2 = (1+√2)/2
18 通项 1/[(n+1)√n+n√(n+1)]
= [(n+1)√n-n√(n+1)]/[n(n+1)^2-(n+1)n^2]
= [(n+1)√n-n√(n+1)]/[n(n+1)]
= 1/√n - 1/√(n+1)
则 原式 = 1-1/√2+1/√2-1/√3+...+1/√2024-1/√2025
= 1-1/√2025 = 1-1/45 = 44/45
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