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Sn = 1/1+√n + 1/2+√2n +.......+1/n+√n2 find limit n→∞ Sn

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


find limit n→∞ Sn

Grade:Upto college level

1 Answers

AKASH GOYAL AskiitiansExpert-IITD
419 Points
10 years ago

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

 

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IITD

 

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