Sn = 1/1+√n + 1/2+√2n +.......+1/n+√n2


find limit n→∞ Sn

3 years ago

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Dear Student


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)


now (1+√x)=t


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


 


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3 years ago

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