1/(√2+1)+1/(√3+√2)+……+1/[√(n+1)+√n]
=(√2-1)/(√2+1)(√2-1)+(√3-√2)/(√3+√2)(√3-√2)+……+[√(n+1)-√n]/[√(n+1)+√n][√(n+1)-√n]
=(√2-1)/(2-1)+(√3-√2)/(3-2)+……+[√(n+1)-√n]/[(n+1)-n]
=(√2-1)/1+(√3-√2)/1+……+[√(n+1)-√n]/1
=√2-1+√3-√2+……+√(n+1)-√n
=√(n+1)-1
1/(√2+1)+1/(√3+√2)+……+1/[√(n+1)+√n]
=(√2-1)/(√2+1)(√2-1)+(√3-√2)/(√3+√2)(√3-√2)+……+[√(n+1)-√n]/[√(n+1)+√n][√(n+1)-√n]
=(√2-1)/(2-1)+(√3-√2)/(3-2)+……+[√(n+1)-√n]/[(n+1)-n]
=(√2-1)/1+(√3-√2)/1+……+[√(n+1)-√n]/1
=√2-1+√3-√2+……+√(n+1)-√n
=√(n+1)-1