IIT-JEE 1981Question 1
Show that $\displaystyle\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}\right)=\log6$.
Definite Integration
Answer$proved: limit = ln 6$
Solution
Concept. The sum has $5n$ terms $\frac1{n+r}$, $r=1,\ldots,5n$. Taking $\frac1n$ out, it is a Riemann sum of $\frac1{1+x}$ on $[0,5]$. Formulas used. (1) $\lim\frac1n\sum_{r=1}^{kn}f\big(\frac rn\big)=\int_0^kf(x)\,dx$Step 1 - Rewrite. $$\sum_{r=1}^{5n}\frac1{n+r}=\frac1n\sum_{r=1}^{5n}\frac{1}{1+\frac rn}.$$ Step 2 - Use (1). The limit is $\int_0^5\frac{dx}{1+x}=\log6-\log1=\log6$. Hence proved. Answer: proved: limit = ln 6