Use an integral to find the following :lim┬(n→∞)( 1/(√(n√n) +1)+1/√n√n+2+…..+1/√n√n+n)?
Use an integral to find the following :lim┬(n→∞)( 1/(√(n√n) +1)+1/√n√n+2+…..+1/√n√n+n)
(lim as n approaches to ∞)
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... lim (n→∞) { 1/ [√n.√(n+1)] + 1/ [√n.√(n+2)] + ··· + 1/ [√n.√(n+n) }
= lim ( n →∞) { Σ (r=1,... n) 1/ [ √n.√(n+r) ] }
= lim ( n→∞) { Σ (r=1,... n) 1/ [ √n.√n. √(1+(r/n)) ] }
= lim ( n→∞) { ( 1/n ) • Σ (r=1,... n) 1/√[ 1+(r/n) ] }
= ∫ [ 0, 1 ] { 1/ √(1+x) } dx
= 2 [ √(1+x) ] from x = 0 to x = 1
= 2 [ √(1+1) - √(1+0) ]
= 2√2 ................................... Ans.
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