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# The questions are in the image attached as a file. Please answer step by step. Thanks a lot.

Antara
45 Points
15 days ago
51.f(r,n)=n-n^2{nn(1+1/n)(1+1/rn)(1+1/r2n).......}n.        (Taking n common from each bracket)
= n-n^2 . nn^2 {1+1/n)(1+1/rn).....}n
Taking log both sides,
ln(f(r,n))=n{ln(1+1/n)+ln(1+1/rn)+ln(1+1/r2n)+......}

Taking lim n->inf both sides
RHS=lim(n->inf) {nln(1+1/n)+nln(1+1/rn)+nln(1+1/r2n)+.......}

Now lim(n->inf) n *ln(1+1/n)   (inf*0 form of limit)
= lim(n->inf)  $\frac{ln(1+1/n))}{1/n}$
=lim(n->inf)    $\frac{1/(1+1/n)*(-1/n^2)}{-1/n^2}$…............(L hospital rule)
= 1
similarly second term=1/r , 3rd term=1/r^2 and so on
so ques 51. ans =e(1+1/r+1/r^2+1/r^3+.......inf)=e3/2  (Putting r=3 and applying sum of inf GP formula)