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Sn = 1/1+√n + 1/2+√2n +.......+1/n+√n2
find limit n→∞ Sn
the sum can be written as ∑r=1r=n 1/(r + √rn)
= ∑r=1r=n 1/n(r/n + √r/n)
now replace 1/n by dx and r/n by x
limit of integral will be r/n
lower limit n tend to ∞ and r=1 will be 0
upper limit for r=n will be 1
it will convert into integral ∫01 dx/(x+√x) = ∫01dx/√x(1+√x)
and dx/2√x = dt
lower limit for x=0 , t=1 andupper limit for x=1, t=2
integral become ∫12 2dt/t= 2(lnt)12= 2ln2=ln4 answer
All the best.
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