Question

Find \(lt_{n\rightarrow\infty}\sum_{a=1}^{n-1}(\frac{ln(1+\frac{a}{n})}{n})\)

(a) ln(2)

(b) ln(4)

(c) 3ln(2)

(d) ^1⁄a

This question was posed to me in quiz.

The doubt is from Indeterminate Forms in division Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (b) ln(4)

The best I can explain: Using limit of sum we have

=\(lt_{n\rightarrow\infty}(\frac{1}{n})\times(ln(1+0)+ln(1+\frac{1}{n})+ln(1+\frac{2}{n})+…+ln(1+\frac{(n-1)}{n}))\)

=\(\int_0^1 ln(1+x)dr=[(x+1)(ln(x+1)-1)]_0^1\)

=(2ln(2)-1)-(ln(1)-1)=2ln(2)

=ln(4)